1 / 32

LP-Based Parameterized Algorithms for Separation Problems

LP-Based Parameterized Algorithms for Separation Problems. D. Lokshtanov , N.S. Narayanaswamy V. Raman , M.S. Ramanujan S. Saurabh. Message of this talk. It was open for quite a while whether Odd Cycle Transversal and Almost 2-Sat are FPT .

gareth
Télécharger la présentation

LP-Based Parameterized Algorithms for Separation Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. LP-Based Parameterized Algorithms for Separation Problems D. Lokshtanov, N.S. Narayanaswamy V. Raman, M.S. Ramanujan S. Saurabh

  2. Message of this talk It was open for quite a while whether Odd Cycle Transversal and Almost 2-Sat are FPT. A simple branching algorithm for Vertex Cover known since the mid 90’s solves both problems in time . Some more work gives .

  3. Results (Above LP) MultiwayCut [CPPW11] (Above LP) Vertex Cover Almost2-SAT Odd Cycle Transversal

  4. How does one get a 4k-LP algorithm? Branching: on both sides k-LP decreases by at least ½. How to improve? Decrease k-LP more.

  5. Multiway Cut In: Graph G, set T of vertices, integer k. Question: such that no component of G\S has at least two vertices of T? FPT by Marx, 04 Faster FPT by Chen et al, 07 Fastest FPT and FPT/k-LP by Cygan et al, 11

  6. Vertex Cover In:G, t Question: such that every edge in G has an endpoint in S? Long story... Here: .

  7. Almost 2-SAT In:2-SAT formula, integer k Question: Can we remove k variables from and make it satisfiable? FPT by Razgon and O’Sullivan, 08 Here: .

  8. Odd Cycle Transversal In:G, k Question: such that G\S is bipartite? FPT: by Reed et al. Here: .

  9. Vertex Cover In:G, t Question: such that every edge in G has an endpoint in S? Minimize Z

  10. Vertex Cover Above LP In:G, t Question: such that every edge in G has an endpoint in S? Running Time:, where LP is the value of the optimum LP solution.

  11. Odd Cycle Transversal  Almost 2-Sat x y x y z z

  12. Almost 2-SAT Vertex Cover/t-LP

  13. Nemhauser Trotter Theorem • There is always an optimal solution to Vertex Cover LP that sets variables to . • For any –solution there is a matching from the 1-vertices to the 0-vertices, saturating the 1-vertices.

  14. Nemhauser Trotter Proof + + - - -

  15. Reduction Rule If exists optimal LP solution that sets xv to 1, then exists optimal vertex cover that selects v. • Remove v from G and decrease t by 1. Correctness follows from Nemhauser Trotter Polynomial time by LP solving.

  16. Branching Pick an edge uv. Solve (G\u, t-1) and (G\v, t-1). since otherwise there is an optimal LP solution for G that sets u to 1. Then

  17. Branching - Analysis LP – t drops by ½ ... in both branches! Total time: Caveat: The reduction does not increase the measure!

  18. Moral Nemhauser Trotter reduction + classic «branch on an edge» gives time algorithm for Vertex Cover and time algorithm for Odd Cycle Transversal and Almost 2-Sat. Can we do better?

  19. Surplus The surplus of a set I is |N(I)| – |I|. The surplus can be negative! In any -LP solution, the total weight is n/2 + surplus(V0)/2. Solving the Vertex Cover LP is equivalent to finding an independent set I of minimum surplus.

  20. Surplus and Reductions If «all ½» is the unique LP optimum then surplus(I) > 0 for all independent sets. Can we say anything meaningful for independent sets of surplus 1? 2? k?

  21. Surplus Branching Lemma Let I be an independent set in G with minimum surplus. There exists an optimal vertex cover C that either containsI or avoidsI.

  22. Surplus Branching Lemma Proof I N(I) R

  23. Branching Rule Find an independent set I of minimum surplus. Solve (G\I, t-|I|) and (G\N(I), t-|N(I)|). LP(G\I) > LP(G) - |I|, since otherwise LP(G) has an optimal solution that sets I to 1. So t-LP drops by at least ½.

  24. Branching Rule Analysis Cont’d Analyzing the (G\N(I), t-N(I)) side: So t-LP drops by at least

  25. Branching Summary The measure k-LP drops by (½, surplus(I)/2). Will see that independent sets of surplus 1 can be reduced in polynomial time! Measure drops by (½,1) giving a time algorithm for Vertex Cover

  26. Reducing Surplus 1 sets. Lemma: If surplus(I) = 1, I has minimum surplus and N(I) is not independent then there exists an optimum vertex cover containing N(I). I N(I) R

  27. Reducing Surplus 1 sets. Reduction Rule: If surplus(I) = 1, I has minimum surplus and N(I) is independent then solve (G’,t-|I|)where G’ is G with N[I] contracted to a single vertex v. I N(I) R

  28. Summary Nemhauser Trotter lets us assume surplus > 0 More rules let us assume surplus > 1 ()* If surplusthen branching yields time for Vertex Cover The correctness of these rules were also proved by NT!

  29. Can we do better? Can get down to by more clever branching rules. Yields for Almost 2-SAT and Odd Cycle Transversal. Should not be the end of the story.

  30. Better OCT? Can we get down to for Odd Cycle Transversal?

  31. LP Branching in other cases I believe many more problems should have FPT algorithms by LP-guided branching. What about ... (Directed) Feedback Vertex Set, parameterized by solution size k?

  32. Thank You!

More Related