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Contributions on Secretary Problems, Independent Sets of Rectangles and Related Problems Jos e A. Soto Doctoral Thesis Defense. Department of Mathematics. M.I.T. April 15th, 2011 Jos e A. Soto - M.I.T. Thesis Defense April 15th, 2011


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Contributions on Secretary Problems, Independent Sets of Rectangles and Related Problems

Jos´ e A. Soto

Doctoral Thesis Defense. Department of Mathematics. M.I.T.

April 15th, 2011

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 1

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Outline

1

Matroid Secretary Problem

2

Jump Number Problem and Independent Sets of Rectangles. (joint work with C. Telha)

3

Symmetric Submodular Function Minimization under Hereditary Constraints.

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Outline

1

Matroid Secretary Problem

2

Jump Number Problem and Independent Sets of Rectangles. (joint work with C. Telha)

3

Symmetric Submodular Function Minimization under Hereditary Constraints.

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MSP: Introduction

Given a matroid.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15 10

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15 10 2

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15 10 2 20

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15 10 2 20 36

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15 10 2 20 36 9

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

3 1 4 15 10 2 20 36 9 5

Given a matroid. Elements’ weights are revealed in certain (random) order.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction

15 20 36 9 3 1 4 10 2 5

Given a matroid. Elements’ weights are revealed in certain (random) order. Want to select independent set of high weight. (In online way / secretary problem setting)

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

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MSP: Introduction (II)

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 1

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 1

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 1

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 1 10

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 1 2 10

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 20 1 2 10

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 20 36 1 2 10

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 20 36 9 1 2 10

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 20 36 9 5 1 2 10

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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MSP: Introduction (II)

3 4 15 20 36 9 5 1 2 10 3 1 4 10 2 5

w(OPT) = 80 w(ALG) = 42

15 20 36 9

Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

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Special Cases

Classical / Multiple choice Hire one person (or at most r). Sell one item to best bidder (or sell r identical items).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 5

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Models

Opponent selects n weights. w1 ≥ w2 ≥ · · · ≥ wn ≥ 0 then The weights are assigned either: adversarially or at random. and independently The presentation order is chosen: adversarially or at random.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

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Models

Opponent selects n weights. w1 ≥ w2 ≥ · · · ≥ wn ≥ 0 then The weights are assigned either: adversarially or at random. and independently The presentation order is chosen: adversarially or at random.

Adv.-Order Random-Assign. Random-Order Adv.-Assign. Random-Order Random-Assign. I.I.D. Weights Adv.-Order Adv.-Assign.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

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Models

(Adv.-Assign. Adv.-Order) Hard: n-competitive ratio

[Babaioff, Immorlica, Kleinberg 07]

Conjecture: O(1)-competitive algorithm for all other models.

Adv.-Order Random-Assign. Random-Order Adv.-Assign. Random-Order Random-Assign. I.I.D. Weights Adv.-Order Adv.-Assign.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

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Models

(Adv.-Assign. Adv.-Order) Hard: n-competitive ratio

[Babaioff, Immorlica, Kleinberg 07]

Conjecture: O(1)-competitive algorithm for all other models. (Adv.-Assign. Random-Order) O(1) for partition, graphic, transversal, laminar.

[L61,D63,K05,BIK07,DP08,KP09,BDGIT09,IW11]

O(log rk(M)) for general matroids [BIK07].

Adv.-Order Random-Assign. Random-Order Random-Assign. I.I.D. Weights Adv.-Order Adv.-Assign. Random-Order Adv.-Assign.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

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Models

(Adv.-Assign. Adv.-Order) Hard: n-competitive ratio

[Babaioff, Immorlica, Kleinberg 07]

Conjecture: O(1)-competitive algorithm for all other models. (Adv.-Assign. Random-Order) O(1) for partition, graphic, transversal, laminar.

[L61,D63,K05,BIK07,DP08,KP09,BDGIT09,IW11]

O(log rk(M)) for general matroids [BIK07]. (Random-Assign. Random-Order) [S11] O(1) for general matroids.

Adv.-Order Random-Assign. Random-Order Adv.-Assign. I.I.D. Weights Adv.-Order Adv.-Assign. Random-Order Random-Assign.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

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Random-Assignment Random-Order.

Data Known Matroid

σ

← − − − −

r.a.

W: w1 ≥ w2 ≥ · · · ≥ wn ≥ 0. Hidden weight list Random assignment. σ: [n] → E. Random order. π: E → {1, . . . , n}. Objective Return an independent set ALG ∈ I such that: Eπ,σ[w(ALG)] ≥ Ω(1) · Eσ[w(OPT)], where OPT is the optimum base of M under assignment σ. (Greedy)

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 7

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Divide and Conquer to get O(1)-competitive algorithm.

For a general matroid M = (E, I): Find matroids Mi = (Ei, Ii) with E = k

i=1 Ei.

1

Mi admits O(1)-competitive algorithm (Easy parts).

2

Union of independent sets in each Mi is independent in M. I(k

i=1 Mi) ⊆ I(M).

(Combine nicely).

3

Optimum in k

i=1 Mi is comparable with

Optimum in M. (Don’t lose much).

M1, E1 M, E M2, E2 Mk, Ek . . .

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 8

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(Easiest matroids): Uniform. [Independent sets = Sets of size ≤ r.] For r = 1: Dynkin’s Algorithm

  • n/e

Observe n/e objects. Accept the first record after that. Top weight is selected w.p. ≥ 1/e.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 9

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(Easiest matroids): Uniform. [Independent sets = Sets of size ≤ r.] For r = 1: Dynkin’s Algorithm

  • n/e

Observe n/e objects. Accept the first record after that. Top weight is selected w.p. ≥ 1/e. General r

  • n/r
  • n/r
  • n/r

· · ·

  • n/r
  • n/r

Divide in r classes and apply Dynkin’s algorithm in each class.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 9

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(Easiest matroids): Uniform. [Independent sets = Sets of size ≤ r.] For r = 1: Dynkin’s Algorithm

  • n/e

Observe n/e objects. Accept the first record after that. Top weight is selected w.p. ≥ 1/e. General r

  • n/r
  • n/r
  • n/r

· · ·

  • n/r
  • n/r

Divide in r classes and apply Dynkin’s algorithm in each class. Each of the r top weights is the best of its class with prob. ≥ (1 − 1/r)r−1 ≥ C > 0. Thus it is selected with prob. ≥ C/e.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 9

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(Easiest matroids): Uniform. [Independent sets = Sets of size ≤ r.] For r = 1: Dynkin’s Algorithm

  • n/e

Observe n/e objects. Accept the first record after that. Top weight is selected w.p. ≥ 1/e. General r

  • n/r
  • n/r
  • n/r

· · ·

  • n/r
  • n/r

Divide in r classes and apply Dynkin’s algorithm in each class. Each of the r top weights is the best of its class with prob. ≥ (1 − 1/r)r−1 ≥ C > 0. Thus it is selected with prob. ≥ C/e. e/C (constant) competitive algorithm.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 9

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(Easy matroids): Uniformly dense matroids are like Uniform A loopless matroid is Uniformly dense if |F| rk(F) ≤ |E| rk(E), for all F = ∅.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 10

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(Easy matroids): Uniformly dense matroids are like Uniform A loopless matroid is Uniformly dense if |F| rk(F) ≤ |E| rk(E), for all F = ∅. Property: Sets of rk(E) elements have almost full rank. E(X:|X|=rk(E))[rk(X)] ≥ rk(E)(1 − 1/e).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 10

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(Easy matroids): Uniformly dense matroids are like Uniform A loopless matroid is Uniformly dense if |F| rk(F) ≤ |E| rk(E), for all F = ∅. Property: Sets of rk(E) elements have almost full rank. E(X:|X|=rk(E))[rk(X)] ≥ rk(E)(1 − 1/e). Algorithm: Simulate e/C-comp. alg. for Uniform Matroids.

  • n/r
  • n/r
  • n/r

· · ·

  • n/r
  • n/r

Try to add each selected weight to the independent set. Selected elements have expected rank ≥ r(1 − 1/e). We recover (1 − 1/e) · C/e fraction of the top r weights.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 10

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Uniformly Dense (sub)matroids That combine nicely

M1, E1 M, E M2, E2 Mk, Ek . . .

Want: Matroids M1, . . . , Mk such that:

1

Each Mi is uniformly dense.

2

If Ii ∈ I(Mi), then I1 ∪ I2 ∪ · · · ∪ Ik ∈ I(M).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 11

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Uniformly Dense (sub)matroids That combine nicely

Procedure. Let E1 be the densest set of M of maximum cardinality. γ(M) := max

F⊆E

|F| rkM(F) = |E1| rkM(E1). M1 = M|E1 is uniformly dense. M∗ = M/E1 has smaller density than M.

M1, E1 M, E M∗, E \ E1

Want: Matroids M1, . . . , Mk such that:

1

Each Mi is uniformly dense.

2

If Ii ∈ I(Mi), then I1 ∪ I2 ∪ · · · ∪ Ik ∈ I(M).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 11

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Uniformly Dense (sub)matroids That combine nicely

Procedure. Let E1 be the densest set of M of maximum cardinality. γ(M) := max

F⊆E

|F| rkM(F) = |E1| rkM(E1). M1 = M|E1 is uniformly dense. M∗ = M/E1 has smaller density than M. Iterate on M∗. . . .

M1, E1 M, E M2, E2 M∗∗, E \ (E1 ∪ E2)

Want: Matroids M1, . . . , Mk such that:

1

Each Mi is uniformly dense.

2

If Ii ∈ I(Mi), then I1 ∪ I2 ∪ · · · ∪ Ik ∈ I(M).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 11

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Uniformly Dense (sub)matroids That combine nicely

Procedure. Let E1 be the densest set of M of maximum cardinality. γ(M) := max

F⊆E

|F| rkM(F) = |E1| rkM(E1). M1 = M|E1 is uniformly dense. M∗ = M/E1 has smaller density than M. Iterate on M∗. . . .

M1, E1 M, E M2, E2 Mk, Ek . . .

Theorem (Principal Partition) [Tomizawa, Narayanan] There exists a partition E = k

i=1 Ei such that

1

Each principal minor Mi = (M/Ei−1)|Ei is uniformly dense.

2

If Ii ∈ I(Mi), then I1 ∪ I2 ∪ · · · ∪ Ik ∈ I(M).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 11

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Algorithm for a General Matroid M

Algorithm

1

Let M1, M2, . . . , Mk be the principal minors.

2

In each Mi use the O(1)-competitive algorithm for uniformly dense matroids to obtain an independent set Ii.

3

Return ALG = I1 ∪ I2 ∪ · · · ∪ Ik. · · ·

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 12

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Algorithm for a General Matroid M

Algorithm

1

Let M1, M2, . . . , Mk be the principal minors.

2

In each Mi use the O(1)-competitive algorithm for uniformly dense matroids to obtain an independent set Ii.

3

Return ALG = I1 ∪ I2 ∪ · · · ∪ Ik. · · · · · · · · ·

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 12

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Algorithm for a General Matroid M

Algorithm

1

Let M1, M2, . . . , Mk be the principal minors.

2

In each Mi use the O(1)-competitive algorithm for uniformly dense matroids to obtain an independent set Ii.

3

Return ALG = I1 ∪ I2 ∪ · · · ∪ Ik. · · · · · · · · · We have: Eπ,σ[w(ALG)] ≥ Ω(1)Eσ[w(OPT Mi)]. Also show Eπ,σ[w(ALG)] ≥ Ω(1)/(1 − 1/e)Eσ[w(OPTM)].

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 12

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Conclusions and Open Problems.

Summary First constant competitive algorithm for Matroid Secretary Problem in Random-Assign. Random-Order Model.

[OG-V] Can use same ideas for Random-Assign. Adv.-Order Model. Algorithm only makes comparisons.

Open Adv.-Assign. Random-Order Model Extend to independent systems beyond matroids.

Adv.-Order Random-Assign. Random-Order Random-Assign. I.I.D. Weights Adv.-Order Adv.-Assign. Random-Order Adv.-Assign. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 13

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Outline

1

Matroid Secretary Problem

2

Jump Number Problem and Independent Sets of Rectangles. (joint work with C. Telha)

3

Symmetric Submodular Function Minimization under Hereditary Constraints.

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Jump Number Problem

a b c d e f

Jumps a ≀ b ≀ cf ≀ d ≀ e 4 jumps a ≀ bd ≀ cf ≀ e 3 jumps

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 14

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Jump Number Problem

a b c d e f

Jumps a ≀ b ≀ cf ≀ d ≀ e 4 jumps a ≀ bd ≀ cf ≀ e 3 jumps Jump number problem for a poset P Find a linear extension (schedule) with minimum number of jumps j(P).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 14

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Jump Number Problem

a b c d e f

Jumps a ≀ b ≀ cf ≀ d ≀ e 4 jumps a ≀ bd ≀ cf ≀ e 3 jumps Jump number problem for a poset P Find a linear extension (schedule) with minimum number of jumps j(P). Properties Comparability invariant. NP-hard even for chordal bipartite graphs. (Every cycle of length ≥ 6 has a chord.)

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 14

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Cross-free Matchings and Biclique Covers.

Cross-free matchings in a bipartite graph G = (A ∪ B, E) Two edges ab and a′b′ cross if ab′ and a′b are also edges. α∗(G) = maximum size of a cross-free matching.

a b c d e f

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 15

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Cross-free Matchings and Biclique Covers.

Cross-free matchings in a bipartite graph G = (A ∪ B, E) Two edges ab and a′b′ cross if ab′ and a′b are also edges. α∗(G) = maximum size of a cross-free matching.

a b c d e f

Fact: For G chordal bipartite. α∗(G) + j(P) = n − 1 Example: a ≀ bd ≀ cf ≀ e

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 15

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Cross-free Matchings and Biclique Covers.

Cross-free matchings in a bipartite graph G = (A ∪ B, E) Two edges ab and a′b′ cross if ab′ and a′b are also edges. α∗(G) = maximum size of a cross-free matching.

a b c d e f

Fact: For G chordal bipartite. α∗(G) + j(P) = n − 1 Example: a ≀ bd ≀ cf ≀ e Biclique Cover in a bipartite graph G = (A ∪ B, E) κ∗(G) = minimum size of a collection of complete bipartite graphs (bicliques) that covers E.

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 15

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Cross-free Matchings and Biclique Covers.

Cross-free matchings in a bipartite graph G = (A ∪ B, E) Two edges ab and a′b′ cross if ab′ and a′b are also edges. α∗(G) = maximum size of a cross-free matching.

a b c d e f

Fact: For G chordal bipartite. α∗(G) + j(P) = n − 1 Example: a ≀ bd ≀ cf ≀ e Biclique Cover in a bipartite graph G = (A ∪ B, E) κ∗(G) = minimum size of a collection of complete bipartite graphs (bicliques) that covers E. α∗(G) ≤ κ∗(G).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 15

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Special Chordal Bipartite Graphs .

Definition (Bicolored 2D-graphs or 2 d.o.r.g.) Given two sets A and B of points in the plane. G(A, B) is the bipartite graph on A ∪ B where ab is an edge if a ∈ A, b ∈ B, ax ≤ bx and ay ≤ by

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 16

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Special Chordal Bipartite Graphs .

Definition (Bicolored 2D-graphs or 2 d.o.r.g.) Given two sets A and B of points in the plane. G(A, B) is the bipartite graph on A ∪ B where ab is an edge if a ∈ A, b ∈ B, ax ≤ bx and ay ≤ by

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 16

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Special Chordal Bipartite Graphs .

Definition (Bicolored 2D-graphs or 2 d.o.r.g.) Given two sets A and B of points in the plane. G(A, B) is the bipartite graph on A ∪ B where ab is an edge if a ∈ A, b ∈ B, ax ≤ bx and ay ≤ by

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 16

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α∗(G(A, B)) and κ∗(G(A, B))

Crossing edges = Overlapping Rectangles Maximal Bicliques = Rectangle Hitting Sets

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 17

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α∗(G(A, B)) and κ∗(G(A, B))

Crossing edges = Overlapping Rectangles Maximal Bicliques = Rectangle Hitting Sets Theorem 1 [ST11] : In a 2 d.o.r.g. with rectangles R α∗ = max. cross-free matching = max. indep. set of R [MIS(R)]. κ∗ = min. biclique cover = min. hitting set of R [MHS(R)].

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 17

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α∗(G(A, B)) and κ∗(G(A, B))

Crossing edges = Overlapping Rectangles Maximal Bicliques = Rectangle Hitting Sets Can replace R by the inclusionwise minimal rectangles R↓. Theorem 1 [ST11] : In a 2 d.o.r.g. with rectangles R α∗ = max. cross-free matching = max. indep. set of R [MIS(R)]. κ∗ = min. biclique cover = min. hitting set of R [MHS(R)].

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 17

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Main results

Theorem 2 [ST11]: In a 2 d.o.r.g. with minimal rectangles R↓ The fractional solution for the natural LP relaxation of MIS(R↓) having minimum weighted area is an integral solution: If P =

  • x ∈ (R+)R↓,
  • R∋q

xR ≤ 1, q ∈ Grid

  • , z∗ = max
  • ✶Tx, x ∈ P
  • .

Then α∗ = z∗ and arg min   

  • R∈R↓

area(R)xR : ✶Tx = z∗, x ∈ P    is integral .

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 18

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Main results

Theorem 2 [ST11]: In a 2 d.o.r.g. with minimal rectangles R↓ The fractional solution for the natural LP relaxation of MIS(R↓) having minimum weighted area is an integral solution: If P =

  • x ∈ (R+)R↓,
  • R∋q

xR ≤ 1, q ∈ Grid

  • , z∗ = max
  • ✶Tx, x ∈ P
  • .

Then α∗ = z∗ and arg min   

  • R∈R↓

area(R)xR : ✶Tx = z∗, x ∈ P    is integral . Theorem 3 [ST11]: For every 2 d.o.r.g. α∗(G(A, B)) = κ∗(G(A, B)).

Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 18

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(sketch) Theorem 3: α∗(G(A, B)) = κ∗(G(A, B)).

H: Intersection graph of R↓. α∗(G(A, B)) = MIS(R↓) = stability number of H. κ∗(G(A, B)) = MHS(R↓) = clique covering number of H.

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SLIDE 68

(sketch) Theorem 3: α∗(G(A, B)) = κ∗(G(A, B)).

H: Intersection graph of R↓. α∗(G(A, B)) = MIS(R↓) = stability number of H. κ∗(G(A, B)) = MHS(R↓) = clique covering number of H. Intersections The only possible intersections in H can be corner-free intersections or corner intersections.

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SLIDE 69

(sketch) Theorem 3: α∗(G(A, B)) = κ∗(G(A, B)).

H: Intersection graph of R↓. α∗(G(A, B)) = MIS(R↓) = stability number of H. κ∗(G(A, B)) = MHS(R↓) = clique covering number of H. Intersections The only possible intersections in H can be corner-free intersections or corner intersections. Perfect Case: If R↓ is such that the only intersections are corner-free-intersection, then its intersection graph H is a comparability graph (perfect). Therefore α∗(G(A, B)) = κ∗(G(A, B)).

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SLIDE 70

(cont.) Theorem 3: MIS(R↓)=MHS(R↓)

General Case:

1

Construct a family K ⊆ R↓ by greedily including (in a certain

  • rder) rectangles in K if they do not form corner-intersection.

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SLIDE 71

(cont.) Theorem 3: MIS(R↓)=MHS(R↓)

General Case:

1

Construct a family K ⊆ R↓ by greedily including (in a certain

  • rder) rectangles in K if they do not form corner-intersection.

2

Since K is a corner-free-intersection family MHS(K)=MIS(K)≤MIS(R↓)≤MHS(R↓).

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SLIDE 72

(cont.) Theorem 3: MIS(R↓)=MHS(R↓)

General Case:

1

Construct a family K ⊆ R↓ by greedily including (in a certain

  • rder) rectangles in K if they do not form corner-intersection.

2

Since K is a corner-free-intersection family MHS(K)=MIS(K)≤MIS(R↓)≤MHS(R↓).

3

Compute a minimum hitting set P of K.

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SLIDE 73

(cont.) Theorem 3: MIS(R↓)=MHS(R↓)

General Case:

1

Construct a family K ⊆ R↓ by greedily including (in a certain

  • rder) rectangles in K if they do not form corner-intersection.

2

Since K is a corner-free-intersection family MHS(K)=MIS(K)≤MIS(R↓)≤MHS(R↓).

3

Compute a minimum hitting set P of K. Swapping procedure. If p, q in P, with px < qx and py < qy s.t. P′ = P \ {p, q} ∪ {(px, qy), (py, qx)} is a hitting set for K then set P ← P′.

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SLIDE 74

(cont.) Theorem 3: MIS(R↓)=MHS(R↓)

General Case:

1

Construct a family K ⊆ R↓ by greedily including (in a certain

  • rder) rectangles in K if they do not form corner-intersection.

2

Since K is a corner-free-intersection family MHS(K)=MIS(K)≤MIS(R↓)≤MHS(R↓).

3

Compute a minimum hitting set P of K. Swapping procedure. If p, q in P, with px < qx and py < qy s.t. P′ = P \ {p, q} ∪ {(px, qy), (py, qx)} is a hitting set for K then set P ← P′. We can show that final P is also a hitting set for R↓.

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SLIDE 75

Conclusions and Other Results

Results in Context

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

? O(n2) O(n9)

Max Cross-Free Matching

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

? O(n2)

Jump Number Min Biclique Cover

* ? ? ? ? ?

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SLIDE 76

Conclusions and Other Results

Results in Context (new results in red)

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

? O(n2) O(n9)

Max Cross-Free Matching

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

? O(n2)

Jump Number Min Biclique Cover

* O(n2.5 log n) O(n2.5 log n)

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SLIDE 77

Conclusions and Other Results

Results in Context (new results in red)

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

? O(n2) O(n9)

Max Cross-Free Matching

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

? O(n2)

Jump Number Min Biclique Cover

* O(n2.5 log n) O(n2.5 log n) O(n2) O(n2)

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SLIDE 78

Conclusions and Other Results

Additional Results

Biconvex Graphs Convex Graphs Two Directional Orthogonal Ray Graphs Chordal Bipartite Graphs Permutation Graphs

NP-hard

Interval Bigraphs Bipartite Permutation Graphs

NP-hard

Max Weight Cross-Free Matching Weighted Jump Number

O(n3) NP-hard ? Show that maximum weight cross-free matching is NP-hard for 2 d.o.r.g. Give O(n3) algorithm for weighted problem in biconvex and convex graphs.

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SLIDE 79

Outline

1

Matroid Secretary Problem

2

Jump Number Problem and Independent Sets of Rectangles. (joint work with C. Telha)

3

Symmetric Submodular Function Minimization under Hereditary Constraints.

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SLIDE 80

SSF Minimization: Introduction

Definitions f : 2V → R is submodular if f(A ∪ B) + f(A ∩ B) ≤ f(A) + f(B), for all A, B ⊆ V f is symmetric if f(A) = f(V \ A), for all A ⊆ V A family I of sets is an independent system if it is closed for inclusion. Problem Find ∅ = X ∗ ∈ I that minimizes f(X) over all X ∈ I.

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SLIDE 81

Examples

Examples Find a minimum unbalanced cut in a (weighted) graph. X V \ X min{|E(X; X)|: 0 = |X| ≤ k}.

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SLIDE 82

Examples

Examples Find a minimum unbalanced cut in a (weighted) graph. X V \ X min{|E(X; X)|: 0 = |X| ≤ k}. Find a nonempty subgraph satisfying an hereditary graph property (e.g. triangle-free, clique, stable-set, planar) minimizing the weights of the edges in its coboundary.

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SLIDE 83

Examples

Examples Find a minimum unbalanced cut in a (weighted) graph. X V \ X min{|E(X; X)|: 0 = |X| ≤ k}. Find a nonempty subgraph satisfying an hereditary graph property (e.g. triangle-free, clique, stable-set, planar) minimizing the weights of the edges in its coboundary. Minimizing a SSF under any combination of upper cardinality / knapsack / matroid constraints.

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SLIDE 84

Results (Old+New)

[Svitkina-Fleischer 08] Minimizing a general submodular function under cardinality constraints is NP-hard to approximate within o(

  • |V|/ log |V|).

[GS10] O(n3)-algorithm for minimizing SSF on independent systems.

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SLIDE 85

Rizzi Functions

Let f be a SSF on V with f(∅) = 0. Define the function d(·, :) on pairs of disjoint subsets of V as d(A, B) = 1 2 (f(A) + f(B) − f(A ∪ B)) . Rizzi A Rizzi bi-set function d(·, :) is any function satisfying

1

Symmetric: d(A, B) = d(B, A).

2

Monotone: d(A, B) ≤ d(A, B ∪ C).

3

Consistent: d(A, C) ≤ d(B, C) ⇒ d(A, B ∪ C) ≤ d(B, A ∪ C). E.g., d(A, B) = |E(A : B)| is a Rizzi bi-set function associated to |δ(·)|.

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SLIDE 86

Pendant Pairs and M.A. order

(s, t) is a pendant pair of d if d({t}, V \ {t}) ≤ d(S, V \ S), for all S separating s and t. v1, . . . , vn is a M.A. order if d(vi, {v1, . . . , vi−1}) ≥ d(vj, {v1, . . . , vi−1}). We get M.A. order by setting v1 arbitrarily and selecting the next vertex as the one with MAX. ADJACENCY to the already selected.

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SLIDE 87

Pendant Pairs and M.A. order

(s, t) is a pendant pair of d if d({t}, V \ {t}) ≤ d(S, V \ S), for all S separating s and t. v1, . . . , vn is a M.A. order if d(vi, {v1, . . . , vi−1}) ≥ d(vj, {v1, . . . , vi−1}). We get M.A. order by setting v1 arbitrarily and selecting the next vertex as the one with MAX. ADJACENCY to the already selected. Lemma [Queyranne, Rizzi] The last two elements (vn−1, vn) of a M.A. order are a pendant pair.

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SLIDE 88

Queyranne’s algorithm

Algorithm to minimize SSF in 2V \ {V, ∅} While |V| ≥ 2,

1

Find (s, t) pendant pair.

2

Add {t} as a candidate for minimum.

3

Fuse s and t as one vertex.

Return the best of the n − 1 candidates.

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SLIDE 89

Queyranne’s algorithm

Algorithm to minimize SSF in 2V \ {V, ∅} While |V| ≥ 2,

1

Find (s, t) pendant pair.

2

Add {t} as a candidate for minimum.

3

Fuse s and t as one vertex.

Return the best of the n − 1 candidates. Remark: If |V| ≥ 3, we can always find a pendant pair avoiding one vertex.

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SLIDE 90

Algorithm for constrained version

A loop of I is a singleton not in I. (Assume I has exactly one loop ℓ). Algorithm While |V| ≥ 3,

1

Find (s, t) pendant pair avoiding ℓ.

2

Add {t} as a candidate for minimum.

3

If {s, t} ∈ I, Fuse s and t as one vertex. Else, Fuse s, t and ℓ as one vertex (call it ℓ).

If |V| = 2, add the only non-loop as a candidate. Return the best candidate.

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SLIDE 91

Conclusions.

Results O(n3)-algorithm for finding all inclusionwise minimal minimizers of a SSF of an independent system I. An algorithm by Nagamochi also solves this problem (and more) in the same time. But our algorithm works for a wider class than Nagamochi’s. Open Characterize functions admitting pendant pairs for all their fusions.

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SLIDE 92

Thank you.

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