Subexponential time algorithms and lower bounds
Description: Subexponential time algorithms and lower bounds for finding path and tree decompositions with few bags Joint work (in progress) with Hans Bodlaender. Jesper Nederlof Technical University Eindhoven Treewidth , , : All containing induce a
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slide1. Subexponential time algorithms and lower bounds for finding path and tree decompositions with few bags Joint work (in progress) with Hans Bodlaender. Jesper Nederlof
Technical University Eindhoven<br>
slide2. 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<br>
slide3. 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<br>
slide4. 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 F G H E<br>
slide5. 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 F G H E<br>
slide6. 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 F G H E<br>
slide7. 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 F G H E G B E<br>
slide8. 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 F G H E G B E<br>
slide9. 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 F G H E G B E Separates A, F and CEH.<br>
slide10. 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 Definition The width of a treedecomposition is                                . The treewidth of a graph is the minimum width among all possible tree decompositions of G.<br>
slide11. Pathwidth ,
                                   ,
               : All       containing    induce a connected subtree. A treedecomposition of graph                       is a pair              where                                      with                and     a tree with vertex set      such that: Definition Definition The width of a treedecomposition is                                . The treewidth of a graph is the minimum width among all possible tree decompositions of G. path path path path path<br>
slide12. Pathwidth Example A B C D F G H E G B E ,
                                   ,
               : All       containing    induce a connected subtree. A treedecomposition of graph                       is a pair              where                                      with                and     a tree with vertex set      such that: Definition path path<br>
slide13. Pathwidth Example A B D G H E G B E ,
                                   ,
               : All       containing    induce a connected subtree. A treedecomposition of graph                       is a pair              where                                      with                and     a tree with vertex set      such that: Definition path path Length: 4<br>
slide14. Bounded length Path decompositions n denotes |V|<br>
slide15. The algorithm<br>
slide16. Naïve branching algorithm Most naïve algorithm you can think of<br>
slide17. Naïve branching algorithm Most naïve algorithm you can think of<br>
slide18. Naïve branching algorithm Most naïve algorithm you can think of A B D G H E<br>
slide19. Naïve branching algorithm Most naïve algorithm you can think of A B D G H E<br>
slide20. Naïve branching algorithm Most naïve algorithm you can think of
Plus a bit of memorization A B D G<br>
slide21. A bit of memorization its incident edges are contained in the right bags<br>
slide22. A bit of memorization<br>
slide23. A bit of memorization<br>
slide24. The lower bound<br>
slide27. Reducing disj.strings to min length pathwidth K=199, length 5<br>
slide28. 1 0 0 1 0 0 0 1 0 1<br>
slide29. 1 0 0 1 0 0 0 1 0 1 8
9
10<br>
slide30. 1 0 0 1 0 0 0 1 0 1 0
1<br>
slide31. Concluding Remarks Upper bound is quite similar to algorithm of Bodlaender and van Rooij (TAPAS’11).
Similar results for minimum length tree decompositions and intervalizing colored graphs.
Thanks for listening!<br>
Technical University Eindhoven<br>
slide2. 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<br>
slide3. 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<br>
slide4. 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 F G H E<br>
slide5. 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 F G H E<br>
slide6. 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 F G H E<br>
slide7. 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 F G H E G B E<br>
slide8. 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 F G H E G B E<br>
slide9. 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 F G H E G B E Separates A, F and CEH.<br>
slide10. 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 Definition The width of a treedecomposition is                                . The treewidth of a graph is the minimum width among all possible tree decompositions of G.<br>
slide11. Pathwidth ,
                                   ,
               : All       containing    induce a connected subtree. A treedecomposition of graph                       is a pair              where                                      with                and     a tree with vertex set      such that: Definition Definition The width of a treedecomposition is                                . The treewidth of a graph is the minimum width among all possible tree decompositions of G. path path path path path<br>
slide12. Pathwidth Example A B C D F G H E G B E ,
                                   ,
               : All       containing    induce a connected subtree. A treedecomposition of graph                       is a pair              where                                      with                and     a tree with vertex set      such that: Definition path path<br>
slide13. Pathwidth Example A B D G H E G B E ,
                                   ,
               : All       containing    induce a connected subtree. A treedecomposition of graph                       is a pair              where                                      with                and     a tree with vertex set      such that: Definition path path Length: 4<br>
slide14. Bounded length Path decompositions n denotes |V|<br>
slide15. The algorithm<br>
slide16. Naïve branching algorithm Most naïve algorithm you can think of<br>
slide17. Naïve branching algorithm Most naïve algorithm you can think of<br>
slide18. Naïve branching algorithm Most naïve algorithm you can think of A B D G H E<br>
slide19. Naïve branching algorithm Most naïve algorithm you can think of A B D G H E<br>
slide20. Naïve branching algorithm Most naïve algorithm you can think of
Plus a bit of memorization A B D G<br>
slide21. A bit of memorization its incident edges are contained in the right bags<br>
slide22. A bit of memorization<br>
slide23. A bit of memorization<br>
slide24. The lower bound<br>
slide27. Reducing disj.strings to min length pathwidth K=199, length 5<br>
slide28. 1 0 0 1 0 0 0 1 0 1<br>
slide29. 1 0 0 1 0 0 0 1 0 1 8
9
10<br>
slide30. 1 0 0 1 0 0 0 1 0 1 0
1<br>
slide31. Concluding Remarks Upper bound is quite similar to algorithm of Bodlaender and van Rooij (TAPAS’11).
Similar results for minimum length tree decompositions and intervalizing colored graphs.
Thanks for listening!<br>