Parameterized approximations via d-skew-symmetric multicut

In this paper we design polynomial time approximation algorithms for several parameterized problems such as Odd Cycle Transversal, Almost 2-SAT, Above Guarantee Vertex Cover and Deletion q-Horn Backdoor Set Detection. Our algorithm proceeds by first reducing the given instance to an instance of the d-Skew-Symmetric Multicut problem, and then computing an approximate solution to this instance. Our algorithm runs in polynomial time and returns a solution whose size is bounded quadratically in the parameter, which in this case is the solution size, thus making it useful as a first step in the design of kernelization algorithms. Our algorithm relies on the properties of a combinatorial object called (L,k)-set, which builds on the notion of (L,k)-components, defined by a subset of the authors to design a linear time FPT algorithm for Odd Cycle Transversal. The main motivation behind the introduction of this object in their work was to replicate in skew-symmetric graphs, the properties of important separators introduced by Marx [2006] which has played a very significant role in several recent parameterized tractability results. Combined with the algorithm of Reed, Smith and Vetta, our algorithm also gives an alternate linear time algorithm for Odd Cycle Transversal. Furthermore, our algorithm significantly improves upon the running time of the earlier parameterized approximation algorithm for Deletion q-Horn Backdoor Set Detection which had an exponential dependence on the parameter; albeit at a small cost in the approximation ratio.