(K, n − k)-max-cut: An o∗(2p)-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 widelystudied 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∗ (2p)-time algorithm, improving upon the previous best O∗ (4p+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, depthsearch 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.