(k, n-k)-Max-Cut: An 𝒪(2)-Time Algorithm and a Polynomial Kernel

Saket Saurabh, Meirav Zehavi

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.