Weighted Coloring in Trees

Julio Araujo 1 Nicolas Nisse 2 Stéphane Pérennes 2
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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 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.
Document type :
Journal articles
Liste complète des métadonnées

Cited literature [14 references]  Display  Hide  Download

https://hal.inria.fr/hal-01109194
Contributor : Nicolas Nisse <>
Submitted on : Sunday, January 25, 2015 - 2:38:27 PM
Last modification on : Friday, April 19, 2019 - 11:12:03 AM
Document(s) archivé(s) le : Friday, September 11, 2015 - 9:15:22 AM

File

journal.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Julio Araujo, Nicolas Nisse, Stéphane Pérennes. Weighted Coloring in Trees. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2014, 28 (4), pp.2029 - 2041. ⟨10.1137/140954167⟩. ⟨hal-01109194⟩

Share

Metrics

Record views

316

Files downloads

142