Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization

Abstract

We show that the NP-complete Feedback Vertex Set problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O ( c k ⋅ m ) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. We extend this to an algorithm enumerating all solutions in O ( d k ⋅ m ) time for a (larger) constant d. As a further result, we present a fixed-parameter algorithm with runtime O ( 2 k ⋅ m 2 ) for the NP-complete Edge Bipartization problem, which asks for at most k edges to remove from a graph to make it bipartite.