An (almost) Linear Time Algorithm For Odd Cycles Transversal

Abstract

We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∊ V(G) with |X| ≤ k such that G – X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number of vertices and the number of edges, respectively, and the function α(m, n) is the inverse of the Ackermann function (see by Tarjan [38]). This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(nm) time algorithm for this problem. Our algorithm also implies the edge version of the problem, i.e, there is an edge set X′ ∊ E(G) such that G – X′ is bipartite. Using this algorithm and the recent result in [16], we give an O(mα(m, n) + n log n) algorithm for the following problem for any fixed k: Input: A graph G and an integer k. Output: Determine whether or not there is a half-integral k disjoint odd cycles packing, i.e, k odd cycles C1, …, Ck in G such that each vertex is on at most two of these odd cycles. This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(n3) time algorithm for this problem. We also give a much simpler and much shorter proof for the following result by Reed [28]. The Erdős-Pósa property holds for the half-integral disjoint odd cycles packing problem. I.e. either G has a half-integral k disjoint odd cycles packing or G has a vertex set X of order at most f(k) such that G – X is bipartite for some function f of k. Note that the Erdős-Pósa property does not hold for odd cycles in general.