Linear Kernels in Linear Time, or How to Save k Colors in O(n2) Steps

This paper examines a parameterized problem that we refer to as n–kGraph Coloring, i.e., the problem of determining whether a graph G with n vertices can be colored using n–k colors. As the main result of this paper, we show that there exists a O(kn2 +k2 + 23.8161k)=O(n2) algorithm for n–kGraph Coloring for each fixed k. The core technique behind this new parameterized algorithm is kernalization via maximum (and certain maximal) matchings. The core technical content of this paper is a near linear-time kernelization algorithm for n–kClique Covering. The near linear-time kernelization algorithm that we present for n–kClique Covering produces a linear size (3k–3) kernel in O(k(n+m)) steps on graphs with n vertices and m edges. The algorithm takes an instance 〈G,k 〉 of Clique Covering that asks whether a graph G can be covered using |V|–k cliques and reduces it to the problem of determining whether a graph G′=(V′,E′) of size ≤ 3k–3 can be covered using |V′| – k′ cliques. We also present a similar near linear-time algorithm that produces a 3k kernel for Vertex Cover. This second kernelization algorithm is the crown reduction rule.