Dual Parameterization of Weighted Coloring

Given a graph G , a proper k - coloring of G is a partition $$c = (S_i)_{i\in [1,k]}$$ c = ( S i ) i ∈ [ 1 , k ] of V ( G ) into k stable sets $$S_1,\ldots , S_{k}$$ S 1 , … , S k . Given a weight function $$w: V(G) \rightarrow {\mathbb {R}}^+$$ w : V ( G ) → R + , the weight of a color $$S_i$$ S i is defined as $$w(i) = \max _{v \in S_i} w(v)$$ w ( i ) = max v ∈ S i w ( v ) and the weight of a coloring c as $$w(c) = \sum _{i=1}^{k}w(i)$$ w ( c ) = ∑ 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  $$\sigma (G,w)$$ σ ( G , w ) , as the minimum weight of a proper coloring of G . The problem of determining  $$\sigma (G,w)$$ 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 $$n^{o(\log n)}$$ 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 $$\sigma (G,w) \le \sum _{v \in V(G)} w(v) - k$$ σ ( G , w ) ≤ ∑ v ∈ 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 $$2^{{\mathcal {O}}(k \log k)} \cdot n^{{\mathcal {O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) , and asked which kernel size can be achieved for the problem. We provide an FPT algorithm in time $$9^k \cdot n^{{\mathcal {O}}(1)}$$ 9 k · n O ( 1 ) , and prove that no algorithm in time $$2^{o(k)} \cdot n^{{\mathcal {O}}(1)}$$ 2 o ( k ) · n O ( 1 ) exists under the ETH . On the other hand, we present a kernel with at most $$(2^{k-1}+1) (k-1)$$ ( 2 k - 1 + 1 ) ( k - 1 ) vertices, and rule out the existence of polynomial kernels unless $$\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}$$ NP ⊆ coNP / 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.