(k,n-k) - Max-Cut: An 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≤|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 O^∗(2^p)-time algorithm, improving upon the previous best 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.