$(k,n-k)$-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.

Appendix

Treewidth: A tree decomposition of a graph G is a pair $(D,\beta )$, where D is a rooted tree and $\beta : V(D)\rightarrow 2^{V(G)}$ is a mapping that satisfies the following conditions.

• For each vertex $v\in V(G)$, the set $\{d\in V(D): v\in \beta (d)\}$ induces a nonempty and connected subtree of D.

• For each edge $\{v,u\}\in E(G)$, there exists $d\in V(D)$ such that $\{v,u\}\subseteq \beta (d)$.

The set $\beta (d)$ is called the bag at d, and the width of $(D,\beta )$ is the size of the largest bag minus one (i.e., $\max _{d\in V(D)}|\beta (d)|-1$). The treewidth of G, $tw_G$, is the minimum width among all possible tree decompositions of G.