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Solving connectivity problems parameterized by treewidth in single exponential time Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk Jesper Nederlof, Dagstuhl seminar Exploiting graph structure to cope with hard problems, 2011 Johan van Rooij and Jakub Onufry Wojtaszszyk Joint work with:

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Outline 1.Definition of treewidth and -graphs. 2.Dynamic programming on -graphs for local problems. 3.Introduction of our main result. 4.The Isolation Lemma. 5. on -graphs using Cut&Count and the Isolation Lemma in time.

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Refer to as a bag.

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E D B A

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E D B A D G B

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E D G B D B A FG D

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E D G B D B A FG D G B E

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E D G B D B A C E B FG D G B E

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition Example A B C D FG H E D G B D B A C E B GH E FG D G B E

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Treewidth : All containing induce a connected subtree A treedecomposition of graph is a pair where with and a tree with vertex set such that: Definition The width of a treedecomposition is. The treewidth of a graph is the minimum width among all possible tree decompositions of G.

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Example (with and ): graphs A simplification of graphs of treewidth ~ made up for this occassion. Definition A graph is a graph with arranged in columns and. So the treewidth of a graph is at most.

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on graphs Theorem[Folklore] on graphs can be solved in time.

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on graphs Theorem[Folklore] on graphs can be solved in time. Proof Idea

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on graphs Theorem[Folklore] on graphs can be solved in time. Proof. For and define as the maximum size of an independent set of such that. Then, Use DP to compute all and return Works for most local problems. Also extends to counting solutions.

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on graphs Definition( ) Given: graph, set terminals and integer Asked: Does there exist of size at most with and connected? Example ( ).

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on graphs Example ( ). Definition( ) Given: graph, set terminals and integer Asked: Does there exist of size at most with and connected?

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on graphs Example ( ). Definition( ) Given: graph, set terminals and integer Asked: Does there exist of size at most with and connected? The straightforward dynamic programming approach needs at least timesteps!

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Cut&Count for Theorem on graphs can be solved in time. Theorem Actually, the orginal and more precise version is: There exists a Monte-Carlo algorithm that given a graph and tree decomposition of width solves in time. The algorithm cannot give false positives and gives false negatives with probability at most.

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The Isolation Lemma Given a set family over a universe and weight function, isolates if there is a unique such that. Definition denotes

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The Isolation Lemma Given a set family over a universe and weight function, isolates if there is a unique such that. Definition denotes Lemma[Mulmuley et al., STOC 87] For every element, choose independently and uniformly at random, then:.

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The Isolation Lemma Given a set family over a universe and weight function, isolates if there is a unique such that. Definition denotes Lemma[Mulmuley et al., STOC 87] For every element, choose independently and uniformly at random, then:.

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The Isolation Lemma Proof Define. So then. This happens with probability at most by union bound. Notice does not depend on. Thus, for a fixed. Now assume are both minimizers, and : Lemma[Mulmuley et al., STOC 87] For every element, choose independently and uniformly at random, then:.

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Definition (The Cut part) Cut&Count for Theorem on graphs can be solved in time. The set of relaxed solutions : the set of solutions :,. Let be an arbitrarily terminal. The set of cut solutions is.

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Example of A cut solution Example ( ).

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Example of A cut solution Example ( ).

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Example of A cut solution Example ( ). Red: in Green: in

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Example of A cut solution Example ( ). Red: in Green: in

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Cut&Count for Observation. Recall the set of cut solutions :. Aha! So suppose we just want to know where there is an even number of solutions, does it help to count ? Proof

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The count part Lemma can be computed in time. Proof sketch For and such that, define as the number of cut solutions of. Write a recurrence relation expressing in terms of. Compute all table entries using dynamic programming and read of the answer from the entries. Corollary can be computed in time.

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Using Isolation lemma Definition. Given a set family and weight, let be Recall, and. 1.Let be chosen independently and uniformly at random 2.For every compute and return iff a one is encountered. Algorithm If is odd for some, we know is non-empty and can safely return. If was non-empty, the isolation lemma tells that (namely the smallest for which ) for some with probability at least.

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Using Isolation lemma Recall, and. 1.Let be chosen independently and uniformly at random 2.For every compute and return iff a one is encountered. If is odd for some, we know is non-empty and can safely return. If was non-empty, the isolation lemma tells that (namely the smallest for which ) for some with probability at least. Algorithm Lemma(recalled) For every element, choose independently and uniformly at random, then:.

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Adding weights Observation. Recall the set of cut solutions :. Proof

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Adding weights Lemma For any, can be computed in time. Proof sketch For, and such that, define as the number of cut solutions of weight of. Write a recurrence relation expressing in terms of. Compute all table entries using dynamic programming and read off the answer from the entries.

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Conclusions Theorem on graphs can be solved in time. In the paper we obtained lots of other results. We ask many open problems, but maybe the best one is: Is it possible to solve connectivity problems parameterized by treewidth in single-exponential time in a deterministic, more intuitive way? For example, is there some structure present that allows us to always ignore many partial solutions? Thanks for listening!

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