Dual parameterization of Weighted Coloring
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Given a graph G, a proper k-coloring of G is a partition c = (S_i)_i∈ [1,k] of V(G) into k stable sets S_1,..., S_k. Given a weight function w: V(G) →R^+, the weight of a color S_i is defined as w(i) = _v ∈ S_i w(v) and the weight of a coloring c as w(c) = ∑_i=1^kw(i). Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair (G,w), denoted by σ(G,w), as the minimum weight of a proper coloring of G. The problem of determining σ(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( n) unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. In this article we provide some positive results for the problem, by considering its so-called dual parameterization: given a vertex-weighted graph (G,w) and an integer k, the question is whether σ(G,w) ≤∑_v ∈ V(G) w(v) - k. We prove that this problem is FPT by providing an algorithm running in time 9^k · n^O(1), and it is easy to see that no algorithm in time 2^o(k)· n^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 we rule out the existence of polynomial kernels unless NP⊆ coNP / poly, even on split graphs with only two different weights. Finally, we identify some classes of graphs on which the problem admits a polynomial kernel, in particular interval graphs and subclasses of split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.
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