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

3. Results (Above LP) Multiway Cut (Above LP) Vertex Cover Almost 2-SAT Odd Cycle Transversal

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

5. Multiway Cut

6. Vertex Cover

7. Almost 2-SAT

8. Odd Cycle Transversal

9. Vertex Cover

10. Vertex Cover Above LP

11. Odd Cycle Transversal  Almost 2-Sat xy z xy z

12. Almost 2-SAT  Vertex Cover/t-LP

13. Nemhauser Trotter Theorem

14. Nemhauser Trotter Proof

15. Reduction Rule If exists optimal LP solution that sets x v 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

17. Branching - Analysis Caveat: The reduction does not increase the measure!

18. Moral Can we do better?

19. 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 contains I or avoids I.

22. Surplus Branching Lemma Proof IN(I) R

23. Branching Rule

24. Branching Rule Analysis Cont’d

25. Branching Summary

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 The correctness of these rules were also proved by NT!

29. Can we do better?

30. Better OCT?

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?