Faster Graph bipartization

摘要

In the Graph bipartization (or Odd Cycle Transversal) problem, the objective is to decide whether a given graph G can be made bipartite by the deletion of k vertices for some given k. The parameterized complexity of Odd Cycle Transversal was resolved in the breakthrough paper of Reed, Smith and Vetta [Operations Research Letters, 2004], who developed an algorithm running in time O(3kkmn). The question of improving the dependence on the input size to linear, which was another long standing open problem in the area, was resolved by Iwata et al. [SICOMP 2016] and Ramanujan and Saurabh [TALG 2017], who presented O(4k(m+n)) and 4kkO(1)(m+n) algorithms respectively. In this paper, we obtain a faster algorithm that runs in time 3kkO(1)(m+n) and hence preserves the linear dependence on the input size while nearly matching the dependence on k incurred by the algorithm of Reed, Smith and Vetta.