# expansions for reductions

Post on 28-Nov-2014

787 views

Embed Size (px)

DESCRIPTION

TRANSCRIPT

- 1. Expansions for Reductions
- 2. Expansions for Reductions Joint work with Fedor Fomin, Daniel Lokshtanov, Geevarghese Philip, and Saket Saurabh University of Bergen University of California, San Diego The Institute of Mathematical Sciences
- 3. Talk Outline
- 4. Talk Outline |N(S)| q|S| q matchings, mutually disjoint in B
- 5. Talk Outline q-Expansion Lemma |N(S)| q|S| q matchings, mutually disjoint in B
- 6. Talk Outline q-Expansion Lemma with applications to Vertex Cover, Feedback Vertex set, and beyond.
- 7. Part 1A Tale of q Matchings
- 8. B AConsider a bipartite graph one of whose parts (say B) is at least twice as big as the other (call this A).
- 9. B AAssume that there are no isolated vertices in B. bleh
- 10. Suppose, further, that for every subset S in A, N(S) is at least twice as large as |S|.
- 11. N(S) SSuppose, further, that for every subset S in A, N(S) is at least twice as large as |S|.
- 12. B Aen there exist two matchings saturating A, bleh
- 13. B Aen there exist two matchings saturating A, bleh
- 14. B Aen there exist two matchings saturating A, and disjoint in B.
- 15. Claim:If N(A) q|A|, then there exists a subset S of A such that: there q matchings saturating the subset S that are vertex-disjoint in B.provided B does not have any isolated vertices.
- 16. Claim:If N(A) q|A|, then there exists a subset S of A such that: there q matchings saturating the subset S that are vertex-disjoint in B.provided B does not have any isolated vertices.
- 17. Claim:If |B| q|A|, then there exists a subset S of A such that: there q matchings saturating the subset S that are vertex-disjoint in B,provided B does not have any isolated vertices.
- 18. Crucially: it turns out that the endpoints of the matchingsin B (the larger set) do not have neighbors outside S.
- 19. B A
- 20. Part 2Two-Expansions and FVS
- 21. Ingredients
- 22. a high-degree vertex, v Ingredients
- 23. a small hitting set, sans va high-degree vertex, v Ingredients
- 24. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Any hitting set whose size is a polynomial function of k.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 25. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.A subset whose removal makes the graph acyclic.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 26. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.A polynomial function of k.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 27. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.At least twice the size of the hitting set.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 28. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Find an approximate hitting set T .When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 29. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.If T does not contain v, we are done.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 30. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Else: v T . Delete T v from G.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 31. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.e only remaining cycles pass through v.When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 32. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Find an optimal cut set for paths from N(v) to N(v).When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 33. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.Why is this cut set small enough?When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 34. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.When is this cut set small enough?When the largest collection of vertex disjoint paths fromN(v) to N(v) is small.
- 35. Given a high-degree vertex v, nding a small hitting set thatdoes not contain v.When is this cut set small enough?When the largest collectio

Recommended

Price 105 CHF Reductions are available for Members of ITU ... Price 105 CHF Reductions are available