Weighted Coloring in Trees

Abstract

International audienceA proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph G as the smallest weight of a proper coloring of G (1997). If vertices of a graph have weight 1, its weighted chromatic number coincides with its chromatic number. Thus, the problem of computing the weighted chromatic number, a.k.a. Max Coloring Problem, is NP-hard in general graphs. It remains NP-hard in some graph classes as bipartite graphs. Approximation algorithms have been designed in several graph classes, in particular, there exists a PTAS for trees. Surprisingly, the time-complexity of computing this parameter in trees is still open. The Exponential Time Hypothesis (ETH) states that 3-SAT cannot be solved in sub-exponen-tial time. We show that, assuming ETH, the best algorithm to compute the weighted chromatic number of n-node trees has time-complexity n Θ(log n) . Our result mainly relies on proving that, when computing an optimal proper weighted coloring of a graph G, it is hard to combine colorings of its connected components

Hyperbolic intersection graphs and (quasi)-polynomial time

“…Quasi-polynomial exact algorithms with matching lower bounds are rare, and so far we know of only a few natural problems in this class. Examples include VC and T D S [33,28,37], and a weighted coloring problem on trees [2]. It seems that problems on intersection graphs in the hyperbolic plane are a natural source of further problems in this class.…”

Ruling out FPT algorithms for Weighted Coloring on forests

“…We would like to mention that, although our reductions use several key ideas introduced by Araújo et al [1], our results are incomparable to those of [1].…”

“…For i ∈ [1,2], to transfer W[i]-hardness from one problem to another, one uses a parameterized reduction, which given an input I = (x, k) of the source problem, computes in time f (k)·|I| c , for some computable function f and a constant c, an equivalent instance I ′ = (x ′ , k ′ ) of the target problem, such that k ′ is bounded by a function depending only on k.…”

“…We denote by (G ′ , w) the instance of Weighted Coloring we are going to construct. We define n = |V (G)| and we fix a bijection β : V (G) → [0, n − 1], which will allow us to define our gadgets depending on integers j ∈ [0, n − 1] corresponding, via β, to the vertices of G. Figure 1, as done in [1]. They will be crucially used in the construction of (G ′ , w).…”

“…We also need the R i -AND gadget, i ∈ [0, 1], depicted in Figure 3, and which is strongly inspired by a similar gadget presented in [1] (called clause tree) corresponding to the logical 'OR'. • For each j ∈ [1,3] and each ℓ ∈ [0, 4k − 1], we introduce a copy of B ℓ and we connect its root to v j .…”