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− k Graph 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(kn+k+2) = O(n) algorithm for n− k Graph 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−k Clique Covering. The near linear-time kernelization algorithm that we present for n−k Clique 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.