Dual Parameterization of Weighted Coloring

Abstract

Given a graph G, a properk-coloring of G is a partition $c = (S_i)_{i\in [1,k]}$ of V(G) into k stable sets $S_1,\ldots , S_{k}$. Given a weight function $w: V(G) \rightarrow {\mathbb {R}}^+$, the weight of a color$S_i$ is defined as $w(i) = \max _{v \in S_i} w(v)$ and the weight of a coloringc as $w(c) = \sum _{i=1}^{k}w(i)$. Guan and Zhu (Inf Process Lett 61(2):77–81, 1997) defined the weighted chromatic number of a pair (G, w), denoted by $\sigma (G,w)$, as the minimum weight of a proper coloring of G. The problem of determining $\sigma (G,w)$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on n-vertex trees in time $n^{o(\log n)}$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. We focus on the so-called dual parameterization of the problem: given a vertex-weighted graph (G, w) and an integer k, is $\sigma (G,w) \le \sum _{v \in V(G)} w(v) - k$? This parameterization has been recently considered by Escoffier (in: Proceedings of the 42nd international workshop on graph-theoretic concepts in computer science (WG). LNCS, vol 9941, pp 50–61, 2016), who provided an FPT algorithm running in time $2^{{\mathcal {O}}(k \log k)} \cdot n^{{\mathcal {O}}(1)}$, and asked which kernel size can be achieved for the problem. We provide an FPT algorithm in time $9^k \cdot n^{{\mathcal {O}}(1)}$, and prove that no algorithm in time $2^{o(k)} \cdot n^{{\mathcal {O}}(1)}$ exists under the ETH. On the other hand, we present a kernel with at most $(2^{k-1}+1) (k-1)$ vertices, and rule out the existence of polynomial kernels unless $\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}$, even on split graphs with only two different weights. Finally, we identify classes of graphs allowing for polynomial kernels, namely interval graphs, comparability graphs, and subclasses of circular-arc and split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.