Spanning trees and the complexity of flood-filling games Kitty - - PowerPoint PPT Presentation

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spanning trees and the complexity of flood filling games

Spanning trees and the complexity of flood-filling games Kitty - - PowerPoint PPT Presentation

Dec 08, 2023 752 likes •1.41k views

Spanning trees and the complexity of flood-filling games Kitty Meeks Alex Scott Mathematical Institute University of Oxford FUN 2012, Venice The original Flood-It game The original Flood-It game The original Flood-It game The original

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Spanning trees and the complexity of flood-filling games

Kitty Meeks Alex Scott

Mathematical Institute University of Oxford

FUN 2012, Venice

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The original Flood-It game

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The original Flood-It game

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The original Flood-It game

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The original Flood-It game

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The original Flood-It game

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The original Flood-It game

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The original Flood-It game

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The original Flood-It game

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The Honey-Bee game

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Generalising to arbitrary graphs

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Generalising to arbitrary graphs

Sources: brighton-hove.dbprimary.com, englishclub.com, picgifs.com

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Existing results concerning Free Flood It

Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph.

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Existing results concerning Free Flood It

Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n × n or 3 × n grids,

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Existing results concerning Free Flood It

Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n × n or 3 × n grids, trees,

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Existing results concerning Free Flood It

Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n × n or 3 × n grids, trees, series-parallel graphs.

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Existing results concerning Free Flood It

Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n × n or 3 × n grids, trees, series-parallel graphs. Free Flood It can be solved in polynomial time when restricted to paths cycles co-comparability graphs

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Existing results concerning Free Flood It

Given a coloured graph G, Free Flood It is the problem of determining the minimum number of moves required to flood G, when we are allowed to make moves anywhere in the graph. Free Flood It is NP-hard when restricted to n × n or 3 × n grids, trees, series-parallel graphs. Free Flood It can be solved in polynomial time when restricted to paths cycles co-comparability graphs

r

if only two colours are used

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Connecting pairs of vertices

G u v

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Connecting pairs of vertices

G u v

Using this fact, we can compute in time O(|V |3|E|c2) the number

f moves required to connect any given pair of vertices in a graph

G = (V , E) coloured with c colours.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Spanning trees

Theorem The number of moves required to flood a coloured graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T.

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Proof of spanning trees result: key step

A

B

The number of moves required to flood T with colour d is at most the sum of the numbers of moves required to flood A and B respectively with colour d.

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This is useless!

In general, a graph has an exponential number of spanning trees.

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This is useless!

In general, a graph has an exponential number of spanning trees. Besides, Free Flood It is still NP-hard even on trees.

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... or is it?

P = NP

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... or is it?

P = NP

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... or is it?

P = NP

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... or is it?

Source: finditinscotland.com

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... or is it?

Source: finditinscotland.com

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... or is it?

A B

The number of moves required to flood G with colour d is at most the sum of the numbers of moves required to flood A and B respectively with colour d.

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... or is it?

H

The number of moves required to flood a subgraph doesn’t increase when we play in a larger graph.

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... or is it?

H

G

The number of moves required to flood a subgraph doesn’t increase when we play in a larger graph.

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Application I: Graphs with polynomially many connected subgraphs

Theorem Free Flood It can be solved in polynomial time on graphs that have only a polynomial number of connected subgraphs.

A

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Application I: Graphs with polynomially many connected subgraphs

Theorem Free Flood It can be solved in polynomial time on graphs that have only a polynomial number of connected subgraphs.

A A1 A2

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Application I: Graphs with polynomially many connected subgraphs

Classes of graphs with only a polynomial number of connected subgraphs include: paths cycles

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Application I: Graphs with polynomially many connected subgraphs

Classes of graphs with only a polynomial number of connected subgraphs include: paths cycles subdivisions of any fixed graph H

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Application II: Connecting k points

Given a coloured graph G and a subset U of at most k vertices, k-Linking Flood It is the problem of determining the number

f moves required to create a single monochromatic component

containing U. Theorem k-Linking Flood It can be solved in time O(|V |k+3|E|c22k) on a graph G = (V , E) coloured with c colours.

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Application II: Connecting k points

G

The number of moves required to connect U is equal to the minimum, taken over all subtrees T of G that contain U, of the number of moves required to flood T.

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Application II: Connecting k points

G

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Application II: Connecting k points

G

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Conclusions

We can analyse flood filling problems by considering only trees.

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Conclusions

We can analyse flood filling problems by considering only trees. This allows us to prove nice complexity results:

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Conclusions

We can analyse flood filling problems by considering only trees. This allows us to prove nice complexity results:

Free Flood It is solvable in polynomial time on graphs with polynomially many connected subgraphs.

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Conclusions

We can analyse flood filling problems by considering only trees. This allows us to prove nice complexity results:

Free Flood It is solvable in polynomial time on graphs with polynomially many connected subgraphs. k-Linking Flood It is solvable in polynomial time on arbitrary graphs (for fixed k).

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Open Problems

Is k-Linking Flood It fixed parameter tractable, parameterised by k?

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Open Problems

Is k-Linking Flood It fixed parameter tractable, parameterised by k? On what other minor-closed classes of trees is Free Flood It solvable in polynomial time?

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Open Problems

Is k-Linking Flood It fixed parameter tractable, parameterised by k? On what other minor-closed classes of trees is Free Flood It solvable in polynomial time? Extremal problems...

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Open Problems

Is k-Linking Flood It fixed parameter tractable, parameterised by k? On what other minor-closed classes of trees is Free Flood It solvable in polynomial time? Extremal problems... Does the Loch Ness Monster exist?

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Thank you