Treewidth reduction for constrained separation and bipartization problems

We present a method for reducing the treewidth of a graph while preserving all the minimal $s-t$ separators. This technique turns out to be very useful for establishing the fixed-parameter tractability of constrained separation and bipartization problems. To demonstrate the power of this technique, we prove the fixed-parameter tractability of a number of well-known separation and bipartization problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set). These results answer a number of open questions in the area of parameterized complexity.Comment: STACS final version of our result. For the complete description of the result please see version

Minimum connected transversals in graphs: New hardness results and tractable cases using the price of connectivity

We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph G is H-free for some graph H if G contains no induced subgraph isomorphic to H. It is known that Connected Vertex Cover is NP-complete even for H-free graphs if H contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if H contains a cycle or claw. In the remaining case H is a linear forest. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for sP2-free graphs for every constant s≥1. For proving these results we use known results on the price of connectivity for vertex cover, feedback vertex set, and odd cycle transversal. This is the first application of the price of connectivity that results in polynomial-time algorithms

Parameterized complexity and approximability of directed odd cycle transversal

A directed odd cycle transversal of a directed graph (digraph) D is a vertex set S that intersects every odd directed cycle of D. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph D and an integer k. The objective is to determine whether there exists a directed odd cycle transversal of D of size at most k. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size k by showing that DOCT does not admit an algorithm with running time f(k)nO(1) unless FPT = W[1]. On the positive side, we give a factor 2 fixed-parameter approximation (FPT approximation) algorithm for the problem. More precisely our algorithm takes as input D and k, runs in time 2O(k)nO(1), and either concludes that D does not have a directed odd cycle transversal of size at most k, or produces a solution of size at most 2k. Finally assuming gap-ETH, we show that there exists an ϵ > 0 such that DOCT does not admit a factor (1 + ϵ) FPT-approximation algorithm