Compression via Matroids

Abstract

The Odd Cycle Transversal problem (OCT) asks whether a given undirected graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result, Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O (4 k kmn) time algorithm for it; this also implies that instances of the problem can be reduced to a so-called problem kernel of size O (4 k ). Since then, the existence of a polynomial kernel for OCT (i.e., a kernelization with size bounded polynomially in k ) has turned into one of the main open questions in the study of kernelization, open even for the special case of planar input graphs. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in k . This represents the first application of matroid theory to kernelization.