Weighted Coloring in Trees

A 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 (1997) defined the weighted chromatic number of a vertex-weighted graph G as the smallest weight of a proper coloring of G. 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-exponential time. We show that, assuming ETH, the best algorithm to compute the weighted chromatic number of n-node trees has time-complexity n O(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

The Proportional Coloring Problem: Optimizing Buffers in Radio Mesh Networks

International audienceIn this paper, we consider a new edge coloring problem to model call scheduling op- timization issues in wireless mesh networks: the proportional coloring. It consists in finding a minimum cost edge coloring of a graph which preserves the propor- tion given by the weights associated to each of its edges. We show that deciding if a weighted graph admits a proportional coloring is pseudo-polynomial while de- termining its proportional chromatic index is NP-hard. We then give lower and upper bounds for this parameter that can be computed in pseudo-polynomial time. We finally identify a class of graphs and a class of weighted graphs for which the proportional chromatic index can be exactly determined

Low-degree graph partitioning via local search with applications to constraint satisfaction, max cut, and coloring

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree \Delta that cut at least a k\Gamma1 k (1 + 1 2\Delta+k\Gamma1 ) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem. These partitions also imply efficient approximations for several problems on weighted bounded-degree graphs. In particular, we improve the best performance ratio for the weighted independent set problem to 3 \Delta+2 , and obtain an efficient algorithm for coloring 3-colorable graphs with at most 3\Delta+2 4 colors