$(k,n-k)$-Max-Cut: An $\mathcal{O}^*(2^p)$-Time Algorithm and a Polynomial Kernel

Abstract

Max-Cut is a well-known classical NP-hard problem. This problem asks whether the vertex-set of a given graph $G=(V,E)$ can be partitioned into two disjoint subsets, A and B, such that there exist at least p edges with one endpoint in A and the other endpoint in B. It is well known that if $p\le |E|/2$, the answer is necessarily positive. A widely-studied variant of particular interest to parameterized complexity, called $(k,n-k)$-Max-Cut, restricts the size of the subset A to be exactly k. For the $(k,n-k)$-Max-Cut problem, we obtain an $\mathcal{O}^*(2^p)$-time algorithm, improving upon the previous best $\mathcal{O}^*(4^{p+o(p)})$-time algorithm, as well as the first polynomial kernel. Our algorithm relies on a delicate combination of methods and notions, including independent sets, depth-search trees, bounded search trees, dynamic programming and treewidth, while our kernel relies on examination of the closed neighborhood of the neighborhood of a certain independent set of the graph G.