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

Message of this talk

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

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.

Multiway Cut

Vertex Cover

Almost 2-SAT

Odd Cycle Transversal

Vertex Cover

Vertex Cover Above LP

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

Almost 2-SAT  Vertex Cover/t-LP

Nemhauser Trotter Theorem

Nemhauser Trotter Proof

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.

Branching

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

Moral Can we do better?

Surplus

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?

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.

Surplus Branching Lemma Proof IN(I) R

Branching Rule

Branching Rule Analysis Cont’d

Branching Summary

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

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

Summary The correctness of these rules were also proved by NT!

Can we do better?

Better OCT?

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?