Firefighting on Trees Beyond Integrality Gaps David Adjiashvili, - - PowerPoint PPT Presentation

Firefighting on Trees Beyond Integrality Gaps

.

David Adjiashvili, Andrea Baggio, Rico Zenklusen

Department of Mathematics ETH Zürich Aussois 2016

Introduction .

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t 1

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t = 0

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t 1

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t = 1

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t 1

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t = 2

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t 1

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.

t = 3

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t 1

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.

t = 4

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t 1

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.

t = 5

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3

Fire Spreading Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

..

r

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t

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t 1

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.

t = 6

3

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t = 0

.

t 1

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t 2

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t 3

.

t 4

.

t 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

..

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t = 0

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t 1

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t 2

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t 3

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t 4

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t 5

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t 6

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t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. protection

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t = 0

.

t 1

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t 2

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t 3

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t 4

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t 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 1

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t

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t = 1

.

t 2

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t 3

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t 4

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t 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 1

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t

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t = 1

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t 2

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t 3

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t 4

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t 5

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t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 2

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t

.

t 1

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t = 2

.

t 3

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t 4

.

t 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 2

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

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t

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t 1

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t = 2

.

t 3

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t 4

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t 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 3

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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r

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t

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t 1

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t 2

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t = 3

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t 4

.

t 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 3

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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r

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t

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t 1

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t 2

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t 3

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t = 4

.

t 5

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t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 4

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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r

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t

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t 1

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t 2

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t 3

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t 4

.

t = 5

.

t 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 5

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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r

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t

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t 1

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t 2

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t 3

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t 4

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t 5

.

t = 6

.

t 1

.

t 2

.

t 3

.

t 4

.

t 5

1

.

.. at t = 6

4

FireFighter Problem - FFP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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r

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t

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t 1

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t 2

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t 3

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t 4

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t 5

.

t = 6

.

t = 1

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t = 2

.

t = 3

.

t = 4

.

t = 5

1

.

.. × time

4

FireFighter Problem - FFP

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..

r

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t

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t 1

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t 2

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t 3

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t 4

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t 5

.

t = 6

.

t = 1

.

t = 2

.

t = 3

.

t = 4

.

t = 5

1

.

.. × time

. GOAL . . Allocate one

.

.. × time as to maximize saved nodes.

4

Resource Minimization for Fire Containment - RMFC

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..

r

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t = 1

.

t = 2

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t = 3

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t = 4

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t = 5

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t = 6

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5

Resource Minimization for Fire Containment - RMFC

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..

r

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t = 1

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t = 2

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t = 3

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t = 4

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t = 5

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t = 6

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terminals

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5

Resource Minimization for Fire Containment - RMFC

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..

r

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t = 1

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t = 2

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t = 3

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t = 4

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t = 5

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t = 6

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. . . . . . . . . . . . . . . . . . .

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save all

. . !

5

Resource Minimization for Fire Containment - RMFC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t = 1

.

t = 2

.

t = 3

.

t = 4

.

t = 5

.

t = 6

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save all

. . !

Using only one

.

.. × time, impossible!

5

Resource Minimization for Fire Containment - RMFC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t = 1

.

t = 2

.

t = 3

.

t = 4

.

t = 5

.

t = 6

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. . . . . . . . . . . . . . . . . . .

.

save all

. . !

With two

.

.. × time, saving all

. . is possible!

5

Resource Minimization for Fire Containment - RMFC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t = 1

.

t = 2

.

t = 3

.

t = 4

.

t = 5

.

t = 6

.

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. . . . . . . . . . . . . . . . . . .

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save all

. . !

. GOAL . . Minimize number of

.

.. × time as to save all

. . .

5

Resource Minimization for Fire Containment - RMFC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t = 1

.

t = 2

.

t = 3

.

t = 4

.

t = 5

.

t = 6

.

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.

save all

. . !

. GOAL . . Minimize number of

.

.. × time as to save all

. . .

5

Previous Results - FireFighter Problem On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . The problem is NP-hard. . Approximation - Hartnell and Li (2000) . . Simple greedy algorithm achieves a 1/2-approximation. . Approximation - Cai, Verbin, and Yang (2008) . . 1

1/ e -approximation (LP based!).

.

• Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012)

. . 1

1/ e -approx. via monotone submodular function

maximization subject to a partition matroid constraint.

6

Previous Results - FireFighter Problem On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . The problem is NP-hard. . Approximation - Hartnell and Li (2000) . . Simple greedy algorithm achieves a 1/2-approximation. . Approximation - Cai, Verbin, and Yang (2008) . . 1

1/ e -approximation (LP based!).

.

• Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012)

. . 1

1/ e -approx. via monotone submodular function

maximization subject to a partition matroid constraint.

6

Previous Results - FireFighter Problem On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . The problem is NP-hard. . Approximation - Hartnell and Li (2000) . . Simple greedy algorithm achieves a 1/2-approximation. . Approximation - Cai, Verbin, and Yang (2008) . . 1

1/ e -approximation (LP based!).

.

• Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012)

. . 1

1/ e -approx. via monotone submodular function

maximization subject to a partition matroid constraint.

6

Previous Results - FireFighter Problem On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . The problem is NP-hard. . Approximation - Hartnell and Li (2000) . . Simple greedy algorithm achieves a 1/2-approximation. . Approximation - Cai, Verbin, and Yang (2008) . . ( 1−1/

e

)

• approximation (LP based!).

.

• Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012)

. . 1

1/ e -approx. via monotone submodular function

maximization subject to a partition matroid constraint.

6

Previous Results - FireFighter Problem On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . The problem is NP-hard. . Approximation - Hartnell and Li (2000) . . Simple greedy algorithm achieves a 1/2-approximation. . Approximation - Cai, Verbin, and Yang (2008) . . ( 1−1/

e

)

• approximation (LP based!).

.

• Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012)

. . ( 1−1/

e

)

• approx. via monotone submodular function

maximization subject to a partition matroid constraint.

6

Previous Results - RMFC On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . NP-hard to approximate within any factor better than 2. O log n -approx. via constant factor approximation for FFP. . Approximation - Chalermsook and Chuzhoy (2010) . . O log n -approximation (LP based!).

7

Previous Results - RMFC On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . NP-hard to approximate within any factor better than 2. O log n -approx. via constant factor approximation for FFP. . Approximation - Chalermsook and Chuzhoy (2010) . . O log n -approximation (LP based!).

7

Previous Results - RMFC On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . NP-hard to approximate within any factor better than 2. O(log n)-approx. via constant factor approximation for FFP. . Approximation - Chalermsook and Chuzhoy (2010) . . O log n -approximation (LP based!).

7

Previous Results - RMFC On Trees

. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) . . NP-hard to approximate within any factor better than 2. O(log n)-approx. via constant factor approximation for FFP. . Approximation - Chalermsook and Chuzhoy (2010) . . O(log∗ n)-approximation (LP based!).

7

Do Integrality Gaps Reflect Approximation Hardness?

Current best algorithms for FFP and RMFC are LP based. FFP 1

1/ e matches integr. gap.

RMFC O log n matches integr. gap up to constant-factor. Answer not clear before! FFP Seen as monotone submodu- lar function max. subject to a partition matroid constraint

(no 1

1/ e

• approx!).

RMFC Similar to the Asymmetric k-center problem (no o log n -approx!).

8

Do Integrality Gaps Reflect Approximation Hardness?

Current best algorithms for FFP and RMFC are LP based. FFP ( 1−1/

e

) matches integr. gap. RMFC O (log∗ n) matches integr. gap up to constant-factor. Answer not clear before! FFP Seen as monotone submodu- lar function max. subject to a partition matroid constraint

(no 1

1/ e

• approx!).

RMFC Similar to the Asymmetric k-center problem (no o log n -approx!).

8

Do Integrality Gaps Reflect Approximation Hardness?

Current best algorithms for FFP and RMFC are LP based. FFP ( 1−1/

e

) matches integr. gap. RMFC O (log∗ n) matches integr. gap up to constant-factor. Answer not clear before! FFP Seen as monotone submodu- lar function max. subject to a partition matroid constraint

(no 1

1/ e

• approx!).

RMFC Similar to the Asymmetric k-center problem (no o log n -approx!).

8

Do Integrality Gaps Reflect Approximation Hardness?

Current best algorithms for FFP and RMFC are LP based. FFP ( 1−1/

e

) matches integr. gap. RMFC O (log∗ n) matches integr. gap up to constant-factor. Answer not clear before! FFP Seen as monotone submodu- lar function max. subject to a partition matroid constraint

(no 1

1/ e

• approx!).

RMFC Similar to the Asymmetric k-center problem (no o log n -approx!).

8

Do Integrality Gaps Reflect Approximation Hardness?

Current best algorithms for FFP and RMFC are LP based. FFP ( 1−1/

e

) matches integr. gap. RMFC O (log∗ n) matches integr. gap up to constant-factor. Answer not clear before! FFP Seen as monotone submodu- lar function max. subject to a partition matroid constraint

(no

( 1−1/

e + ϵ

)

• approx!).

RMFC Similar to the Asymmetric k-center problem (no o log n -approx!).

8

Do Integrality Gaps Reflect Approximation Hardness?

Current best algorithms for FFP and RMFC are LP based. FFP ( 1−1/

e

) matches integr. gap. RMFC O (log∗ n) matches integr. gap up to constant-factor. Answer not clear before! FFP Seen as monotone submodu- lar function max. subject to a partition matroid constraint

(no

( 1−1/

e + ϵ

)

• approx!).

RMFC Similar to the Asymmetric k-center problem (no o(log∗ n)-approx!).

8

Our Contributions

. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . PTAS for the FireFighter Problem on trees. . Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . O 1 -approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps.

9

Our Contributions

. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . PTAS for the FireFighter Problem on trees. . Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . O 1 -approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps.

9

Our Contributions

. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . PTAS for the FireFighter Problem on trees. . Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . O(1)-approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps.

9

Our Contributions

. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . PTAS for the FireFighter Problem on trees. . Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . O(1)-approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps.

9

Our Contributions

. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . PTAS for the FireFighter Problem on trees. . Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . O(1)-approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps.

9

PTAS For The FireFighter Problem .

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

V nodes r xu 1 if

.

..

• w.

Vt layers t cu subtree of u Pv path r v LP max

u V cu xu

s t

w P

vxw

1 v leaves

w Vtxw

t t 1 L xu 0 1 u V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

V nodes r xu 1 if

.

..

• w.

Vt layers t cu subtree of u Pv path r v LP max

u V cu xu

s t

w P

vxw

1 v leaves

w Vtxw

t t 1 L xu 0 1 u V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

V = nodes \ {r} xu 1 if

.

..

• w.

Vt layers t cu subtree of u Pv path r v LP max

u V cu xu

s t

w P

vxw

1 v leaves

w Vtxw

t t 1 L xu 0 1 u V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

.

u V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt layers t cu subtree of u Pv path r v LP max

u V cu xu

s t

w P

vxw

1 v leaves

w Vtxw

t t 1 L xu 0 1 u V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

.

u V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt layers t cu subtree of u Pv path r v LP max

u V cu xu

• s. t.

w P

vxw

1 v leaves

w Vtxw

t t 1 L xu ∈ {0, 1} ∀ u ∈V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=5

.

t=L

.

u

.

t=4

.

u V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt = layers ≤ t cu subtree of u Pv path r v LP max

u V cu xu

• s. t.

w P

vxw

1 v leaves ∑

w∈ Vtxw ≤ t

∀ t = 1, . . . , L xu ∈ {0, 1} ∀ u ∈V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=5

.

t=L

.

u

.

t=4

.

u

.

u V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt = layers ≤ t cu = |subtree of u | Pv path r v LP max ∑

u∈ V cu xu

• s. t.

w P

vxw

1 v leaves ∑

w∈ Vtxw ≤ t

∀ t = 1, . . . , L xu ∈ {0, 1} ∀ u ∈V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=5

.

t=L

.

u

.

t=4

.

u

.

u

.

v V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt = layers ≤ t cu = |subtree of u | Pv = path r → v LP max ∑

u∈ V cu xu

• s. t.

∑

w∈P

vxw ≤ 1

∀ v ∈ leaves ∑

w∈ Vtxw ≤ t

∀ t = 1, . . . , L xu ∈ {0, 1} ∀ u ∈V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt = layers ≤ t cu = |subtree of u | Pv = path r → v (LP ) max ∑

u∈ V cu xu

• s. t.

∑

w∈P

vxw ≤ 1

∀ v ∈ leaves ∑

w∈ Vtxw ≤ t

∀ t = 1, . . . , L xu ∈ [0, 1] ∀ u ∈V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt = layers ≤ t cu = |subtree of u | Pv = path r → v (LP ) max ∑

u∈ V cu xu

• s. t.

∑

w∈P

vxw ≤ 1

∀ v ∈ leaves ∑

w∈ Vtxw ≤ t

∀ t = 1, . . . , L xu ∈ [0, 1] ∀ u ∈V

11

The Linear Program

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=L

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

V = nodes \ {r} xu = { 1 if

.

..

• w.

Vt = layers ≤ t cu = |subtree of u | Pv = path r → v (LP ) max ∑

u∈ V cu xu

• s. t.

∑

w∈P

vxw ≤ 1

∀ v ∈ leaves ∑

w∈ Vtxw ≤ t

∀ t = 1, . . . , L xu ∈ [0, 1] ∀ u ∈V

11

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

Let x be a vertex sol. of LP . . . A node u is loose w.r.t. x if

• xu
• w P

u xw

1 . . A node u is tight w.r.t. x if

• xu
• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

. . . . . . .

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x if

• xu
• w P

u xw

1 . . A node u is tight w.r.t. x if

• xu
• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

. . . . . . .

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• xu
• w P

u xw

1 . . A node u is tight w.r.t. x if

• xu
• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• x∗

u > 0

• w P

u xw

1 . . A node u is tight w.r.t. x if

• xu
• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

.

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw < 1

. . A node u is tight w.r.t. x if

• xu
• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

. . . . . . .

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw < 1

. . A node u is tight w.r.t. x∗ if

• xu
• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw < 1

. . A node u is tight w.r.t. x∗ if

• x∗

u > 0

• w P

u xw

1 . PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

.

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw < 1

. . A node u is tight w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw = 1

. PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

do nothing push fraction down to some

. .

• 2. return integral part

12

Loose and Tight Nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.25

.

.25

.

.5

.

.75

.

.25

.

.25

. . . . . . . .

Let x∗ be a vertex sol. of (LP ). . . A node u is loose w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw < 1

. . A node u is tight w.r.t. x∗ if

• x∗

u > 0

• ∑

w∈P

u xw = 1

. PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

{ do nothing push fraction down to some

. .

• 2. return integral part

12

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most OPT.

. .

. have to become few

and reallocation light!

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

⇐

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

⇐

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

⇐

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

⇐

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

PLAN - Reallocate Loose Nodes

We want loss from reallocating

. . at most ϵOPT.

. .

. have to become few

and reallocation light!

⇐

• tree transformations
• ad-hoc guessing

Bound on number of

. . :

. Sparsity Lemma . . Number of loose nodes at most L (depth of tree). We need to reduce L!

13

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

2

.

4

cu push down

.

..

l 2i, i 0 1 erase connect . .

• Optimal value LPcomp

1/2 optimal value LPorig .

• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

2

.

4

cu push down

.

..

l 2i, i 0 1 erase connect . .

• Optimal value LPcomp

1/2 optimal value LPorig .

• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

2

.

4

cu push down

.

..

l 2i, i 0 1 erase connect . .

• Optimal value LPcomp

1/2 optimal value LPorig .

• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

×2

.

×4

cu push down

.

..

l=2i, i=0,1,... erase connect . .

• Optimal value LPcomp

1/2 optimal value LPorig .

• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

×2

.

×4

cu push down

.

..

l=2i, i=0,1,... erase connect . .

• Optimal value LPcomp

1/2 optimal value LPorig .

• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

×2

.

×4

cu push down

.

..

l=2i, i=0,1,... erase connect . .

• Optimal value LPcomp

1/2 optimal value LPorig .

• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

×2

.

×4

cu push down

.

..

l=2i, i=0,1,... erase connect . .

• Optimal value (LPcomp) ≥ 1/2 optimal value (LPorig).
• Compressed tree has

log L layers.

• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

×2

.

×4

cu push down

.

..

l=2i, i=0,1,... erase connect . .

• Optimal value (LPcomp) ≥ 1/2 optimal value (LPorig).
• Compressed tree has ≈ log(L) layers.
• Poly-time transformation.

14

Compression - Reducing Depth L

. . . 28 . . . 23 . . . 10 . . . 9 . . . 1 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 12 . . . 1 . . . 10 . . . 1 . . . 7 . . . 1 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 4 . . 1 . . . 2 . . . 1 . . . 1 . . . 22 . . . 2 . . . 1 . . . 19 . . . 18 . . . 4 . . . 2 . . . 1 . . . 1 . . . 13 . . . 12 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 2 . . . 1 . . . 1 . . . 25 . . . 7 . . . 6 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 1 . . . 16 . . . 15 . . . 3 . . . 2 . . . 1 . . . 11 . . . 9 . . . 8 . . . 3 . . . 1 . . . 1 . . . 1 . . . 3 . . . 1 . . . 1 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

t=6

.

t=7

.

t=8

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

×2

.

×4

cu push down

.

..

l=2i, i=0,1,... erase connect . .

• Optimal value (LPcomp) ≥ 1/2 optimal value (LPorig).
• Compressed tree has ≈ log(L) layers.
• Poly-time transformation.

14

δ-compression

Previously, we selected l 2i, i 0 1 What if we select l 1

n , n

0 1 , 0 1 ? . Lemma . .

• Optimal value LPcomp

1

• ptimal value LPorig .
• Compressed tree has O log L/

layers.

• Poly-time transformation.

15

δ-compression

Previously, we selected l = 2i, i = 0, 1, . . . What if we select l 1

n , n

0 1 , 0 1 ? . Lemma . .

• Optimal value LPcomp

1

• ptimal value LPorig .
• Compressed tree has O log L/

layers.

• Poly-time transformation.

15

δ-compression

Previously, we selected l = 2i, i = 0, 1, . . . What if we select l = ⌊(1 + δ)n⌋, n = 0, 1, . . . , δ ∈ ]0, 1[? . Lemma . .

• Optimal value LPcomp

1

• ptimal value LPorig .
• Compressed tree has O log L/

layers.

• Poly-time transformation.

15

δ-compression

Previously, we selected l = 2i, i = 0, 1, . . . What if we select l = ⌊(1 + δ)n⌋, n = 0, 1, . . . , δ ∈ ]0, 1[? . Lemma . .

• Optimal value (LPcomp) ≥ (1 − δ) optimal value (LPorig).
• Compressed tree has O log L/

layers.

• Poly-time transformation.

15

δ-compression

Previously, we selected l = 2i, i = 0, 1, . . . What if we select l = ⌊(1 + δ)n⌋, n = 0, 1, . . . , δ ∈ ]0, 1[? . Lemma . .

• Optimal value (LPcomp) ≥ (1 − δ) optimal value (LPorig).
• Compressed tree has O

(log L/δ ) layers.

• Poly-time transformation.

15

δ-compression

Previously, we selected l = 2i, i = 0, 1, . . . What if we select l = ⌊(1 + δ)n⌋, n = 0, 1, . . . , δ ∈ ]0, 1[? . Lemma . .

• Optimal value (LPcomp) ≥ (1 − δ) optimal value (LPorig).
• Compressed tree has O

(log L/δ ) layers.

• Poly-time transformation.

15

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.8

.

.2

.

.9

.

.1

.

.1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.8

.

.2

.

.9

.

.1

.

.1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

. PLAN: reallocate

. .

and discard fractions . .

• 1. for each

. . :

{ do nothing push fraction down to some

. .

• 2. return integral part

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.8

.

.2

.

.9

.

.1

.

.1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

. PLAN: reallocate

. .

and discard fractions . .

• 1. for each

. . :

{ do nothing push fraction down to some

. .

• 2. return integral part

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

.8

.

.2

.

.9

.

.1

.

.1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

. PLAN: reallocate

. .

and discard fractions . .

• 1. for each

. . :

{ do nothing push fraction down to some

. .

• 2. return integral part

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

. PLAN: reallocate

. .

and discard fractions . .

• 1. for each

. . :

{ do nothing push fraction down to some

. .

• 2. return integral part

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

. PLAN: reallocate

. . and discard fractions

. .

• 1. for each

. . :

{ do nothing push fraction down to some

. .

• 2. return integral part

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

How to enforce little loss from reallocation?

• either

. . covers little

• or (

. . . ) loses little

Loss measure number of

. nodes.

Frac.

. . always covered by . . . always covered by . . .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

How to enforce little loss from reallocation?

• either

. . covers little

• or (

. . . ) loses little

Loss measure number of

. nodes.

Frac.

. . always covered by . . . always covered by . . .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

How to enforce little loss from reallocation?

• either

. . covers little

• or (

. . → . . ) loses little

Loss measure number of

. nodes.

Frac.

. . always covered by . . . always covered by . . .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

How to enforce little loss from reallocation?

• either

. . covers little

• or (

. . → . . ) loses little

Loss measure = number of

. nodes.

Frac.

. . always covered by . . . always covered by . . .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

How to enforce little loss from reallocation?

• either

. . covers little

• or (

. . → . . ) loses little

Loss measure = number of

. nodes.

Frac.

. . always covered by . . . always covered by . . .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

.8

.

.2

. .

.1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

How to enforce little loss from reallocation?

• either

. . covers little

• or (

. . → . . ) loses little

Loss measure = number of

. nodes.

Frac.

. . always covered by . . ⇒ . always covered by . . .

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

IDEA: •

. . has to be “almost” a leaf

• .

. has to be “almost” like some . .

SOLUTION: force

. . and . . to appear only in delimited regions

by guessing a critical set of nodes w.r.t. optimal solution.

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

IDEA: •

. . has to be “almost” a leaf

• .

. has to be “almost” like some . .

SOLUTION: force

. . and . . to appear only in delimited regions

by guessing a critical set of nodes w.r.t. optimal solution.

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

IDEA: •

. . has to be “almost” a leaf

• .

. has to be “almost” like some . .

SOLUTION: force

. . and . . to appear only in delimited regions

by guessing a critical set of nodes w.r.t. optimal solution.

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

. Guessing . . Assume opt is an optimal integral solution, for any

. .

u : add

w P

u xw

1 to LP if u saved by opt add

w P

u xw

0 to LP if u burning in opt

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

. Guessing . . Assume opt is an optimal integral solution, for any

. .

u : add

w P

u xw

1 to LP if u saved by opt add

w P

u xw

0 to LP if u burning in opt

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

. Guessing . . Assume opt is an optimal integral solution, for any

. .

u : { add ∑

w∈P

u xw = 1 to (LP )

if u saved by opt add ∑

w∈P

u xw = 0 to (LP )

if u burning in opt

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

Solve (LP ) updated with guessed contraints

• bserve: • either

. . does not have . . below

• or

. . covers some . . of the same component

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. .

loose

. .

tight

. .

loss

.

critical

. .

Solve (LP ) updated with guessed contraints

• bserve: • either

. . does not have . . below

• or

. . covers some . . of the same component

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

.

loose

. .

tight

. .

loss

.

critical

. .

How to place

. . as to contain loss?

• .

. have to isolate small components

• any nearest common ancestor of

. . has to be . .

. . loss by reallocation bounded by size of component

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

How to place

. . as to contain loss?

• .

. have to isolate small components

• any nearest common ancestor of

. . has to be . .

. . loss by reallocation bounded by size of component

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

How to place

. . as to contain loss?

• .

. have to isolate small components

• any nearest common ancestor of

. . has to be . .

. . loss by reallocation bounded by size of component

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

How to place

. . as to contain loss?

• .

. have to isolate small components

• any nearest common ancestor of

. . has to be . .

. . ⇒ loss by reallocation bounded by size of component

16

Critical Nodes - Bounding Loss From Singular Reallocation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

r

.

8

.

2

.

9

.

1

.

1

.

8

.

2

. .

1

.

.

1

. . .

loose

. .

tight

. .

loss

.

critical

. .

. ROUNDING: reallocate

. . and discard fractions

. .

• 1. for each

. . :

{ do nothing if no

. . below

push down to the

. . covering . .

• 2. return integral part

16

k -Critical Nodes

Let loss for each reallocation k. (

. . has to separate components of size at most k.)

Recall: • there are at most O log L reallocations

• we need total loss

OPT We need k

OPT/ O log L .

Number of

. . is O

V OPT/ O log L

. However, guessing costs 2

. . .

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O log L reallocations

• we need total loss

OPT We need k

OPT/ O log L .

Number of

. . is O

V OPT/ O log L

. However, guessing costs 2

. . .

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O log L reallocations

• we need total loss

OPT We need k

OPT/ O log L .

Number of

. . is O

V OPT/ O log L

. However, guessing costs 2

. . .

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O(log L) reallocations

• we need total loss ≤ ϵOPT

We need k

OPT/ O log L .

Number of

. . is O

V OPT/ O log L

. However, guessing costs 2

. . .

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O(log L) reallocations

• we need total loss ≤ ϵOPT

We need k ≤ ϵOPT/

O(log L).

Number of

. . is O

V OPT/ O log L

. However, guessing costs 2

. . .

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O(log L) reallocations

• we need total loss ≤ ϵOPT

We need k ≤ ϵOPT/

O(log L).

Number of

. . is O

(

| V | ϵOPT/ O(log L)

) . However, guessing costs 2

. . .

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O(log L) reallocations

• we need total loss ≤ ϵOPT

We need k ≤ ϵOPT/

O(log L).

Number of

. . is O

(

| V | ϵOPT/ O(log L)

) . However, guessing costs = 2|

. . |.

For poly-size enumeration, we need OPT V !

17

k -Critical Nodes

Let loss for each reallocation ≤ k. (

. . has to separate components of size at most k.)

Recall: • there are at most O(log L) reallocations

• we need total loss ≤ ϵOPT

We need k ≤ ϵOPT/

O(log L).

Number of

. . is O

(

| V | ϵOPT/ O(log L)

) . However, guessing costs = 2|

. . |.

For poly-size enumeration, we need OPT ≈ | V |!

17

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

. Greedy Algorithm . . From the first to the last layer, place

.

.. at the heaviest remaining subtree.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

1

.

.. × time

. Greedy Algorithm . . From the first to the last layer, place

.

.. at the heaviest remaining subtree.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

. Greedy Pruning . .

• 1. From the first to the last layer,

place

.

.. at the heaviest remaining subtrees.

• 2. Remove all the nodes that are not saved

but leave the path from

.

.. to r.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

. Greedy Pruning . .

• 1. From the first to the last layer,

place λ

.

.. at the heaviest remaining subtrees.

• 2. Remove all the nodes that are not saved

but leave the path from

.

.. to r.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

λ = 2

.

.. × time

. Greedy Pruning . .

• 1. From the first to the last layer,

place λ

.

.. at the heaviest remaining subtrees.

• 2. Remove all the nodes that are not saved

but leave the path from

.

.. to r.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

λ = 2

.

.. × time

. Greedy Pruning . .

• 1. From the first to the last layer,

place λ

.

.. at the heaviest remaining subtrees.

• 2. Remove all the nodes that are not saved

but leave the path from

.

.. to r.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

λ = 2

.

.. × time

. Lemma . .

• Optimal value in pruned tree

1/ size of pruned tree.

• Optimal value in pruned tree

1

1/

• ptimal value in original tree.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

λ = 2

.

.. × time

. Lemma . .

• Optimal value in pruned tree ≥ 1/λ size of pruned tree.
• Optimal value in pruned tree

1

1/

• ptimal value in original tree.

18

λ-Greedy Pruning

. . . . 9 . . . 7 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 12 . . . 11 . . . 4 . . . 3 . . . 1 . . . 1 . . . 6 . . . 1 . . . 4 . . . 1 . . . 1 . . . 1 . . . 4 . . . 2 . . . 1 . . . 1 . . . 8 . . . 6 . . . 1 . . . 4 . . . 1 . . . 2 . . . 1 . . . 1 . . . 11 . . . 2 . . . 1 . . . 8 . . . 6 . . . 3 . . . 1 . . . 1 . . . 2 . . . 1 . . . 1 . . . 2 . . . 1 . . . 7 . . . 6 . . . 5 . . . 2 . . . 1 . . . 2 . . . 1 . . . 8 . . . 1 . . . 6 . . . 5 . . . 1 . . . 3 . . . 1 . . . 1 . . . 10 . . . 1 . . . 8 . . . 1 . . . 6 . . . 2 . . . 1 . . . 3 . . . 1 . . . 1 . . . 9 . . . 3 . . . 1 . . . 1 . . . 5 . . . 4 . . . 1 . . . 2 . . . 1

..

r

.

t=1

.

t=2

.

t=3

.

t=4

.

t=5

.

. 12

.

. 8

.

. 6

.

. 3

.

. 1

.

. 12

.

. 11

.

. 7

.

. 8

.

. 5

.

. 5

.

. 2

.

. 2 .

λ = 2

.

.. × time

. Lemma . .

• Optimal value in pruned tree ≥ 1/λ size of pruned tree.
• Optimal value in pruned tree ≥

≥ ( 1 − 1/λ )

• ptimal value in original tree.

18

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, DO:

• 1. apply
• greedy pruning
• 2. apply -compression
• 3. guess correctly k -critical nodes
• 4. solve LP

localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply
• greedy pruning
• 2. apply -compression
• 3. guess correctly k -critical nodes
• 4. solve LP

localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply
• greedy pruning
• 2. apply -compression
• 3. guess correctly k -critical nodes
• 4. solve LP

localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply λ-greedy pruning
• 2. apply -compression
• 3. guess correctly k -critical nodes
• 4. solve LP

localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply λ-greedy pruning
• 2. apply δ-compression
• 3. guess correctly k -critical nodes
• 4. solve LP

localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply λ-greedy pruning
• 2. apply δ-compression
• 3. guess correctly k -critical nodes
• 4. solve LP

localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply λ-greedy pruning
• 2. apply δ-compression
• 3. guess correctly k -critical nodes
• 4. solve (LP ) → localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply λ-greedy pruning
• 2. apply δ-compression
• 3. guess correctly k -critical nodes
• 4. solve (LP ) → localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

The Algorithm

. Algorithm - PTAS for FFP . . INPUT: tree of size n, ϵ > 0 DO:

• 1. apply λ-greedy pruning
• 2. apply δ-compression
• 3. guess correctly k -critical nodes
• 4. solve (LP ) → localize

. . and . .

• 5. reallocate

. .

RETURN: integral part

19

Conclusions .

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O 1 -approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of -compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O(1)-approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of -compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O(1)-approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of -compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O(1)-approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of -compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O(1)-approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of δ-compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O(1)-approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of δ-compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21

Conclusions

. Theorem . . PTAS for the FireFighter Problem on trees. . Theorem . . O(1)-approximation for RMFC on trees. . . We use:

• LP solution structure
• variant of δ-compression
• iterative enumeration

OPEN: Is there a 2-approximation?

21