On the Kernelization Complexity of Problems on Graphs without Long Odd Cycles

Abstract

Several NP-hard problems, like Maximum Independent Set, Coloring, and Max-Cut are polynomial time solvable on bipartite graphs. An equivalent characterization of bipartite graphs is that it is the set of all graphs that do not contain any odd length cycle. Thus, a natural question here is what happens to the complexity of these problems if we know that the length of the longest odd cycle is bounded by k? Let ${\mathcal O}_k$ denote the set of all graphs G such that the length of the longest odd cycle is upper bounded by k. Hsu, Ikura and Nemhauser [Math. Programming, 1981] studied the effect of avoiding long odd cycle for the Maximum Independent Set problem and showed that a maximum sized independent set on a graph $G\in{\mathcal O}_k$ on n vertices can be found in time n ^O(k). Later, Grötschel and Nemhauser [Math. Programming, 1984] did a similar study for Max-Cut and obtained an algorithm with running time n ^O(k) on a graph $G\in{\mathcal O}_k$ on n vertices.

In this paper, we revisit these problems together with q -Coloring and observe that all of these problems admit algorithms with running time O(c ^k n ^O(1)) on a graph $G\in{\mathcal O}_k$ on n vertices. Thus, showing that all these problems are fixed parameter tractable when parameterized by the length of the longest odd cycle of the input graph. However, following the recent trend in parameterized complexity, we also study the kernelization complexity of these problems. We show that Maximum Independent Set, q -Coloring for some fixed q ≥ 3 and Max-Cut do not admit a polynomial kernel unless $\mbox{\sc co-NP} \subseteq \mbox{\sc NP}/\mbox{\textrm{poly}}$, when parameterized by k, the length of the longest odd cycle.