Ruling out FPT algorithms for Weighted Coloring on forests - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Electronic Notes in Discrete Mathematics Année : 2017

Ruling out FPT algorithms for Weighted Coloring on forests

Julien Baste
Ignasi Sau

Résumé

Given a graph G, a proper k-coloring of G is a partition c = (S i) i∈[0,k−1] of V (G) into k stable sets S 0 ,. .. , S k−1. Given a weight function w : V (G) → R + , the weight of a color S i is dened as w(i) = max v∈S i w(v) and the weight of a coloring c as w(c) = k−1 i=0 w(i). Guan and Zhu [Inf. Process. Lett., 1997] dened the weighted chromatic number of a pair (G, w), denoted by σ(G, w), as the minimum weight of a proper coloring of G. For a positive integer r, they also dened σ(G, w; r) as the minimum of w(c) among all proper r-colorings c of G. The complexity of determining σ(G, w) when G is a tree was open for almost 20 years, until Araújo et al. [SIAM J. Discrete Math., 2014] recently proved that the problem cannot be solved in time n o(log n) on n-vertex trees unless the Exponential Time Hypothesis (ETH) fails. The objective of this article is to provide hardness results for computing σ(G, w) and σ(G, w; r) when G is a tree or a forest, relying on complexity assumptions weaker than the ETH. Namely, we study the problem from the viewpoint of parameterized complexity, and we assume the weaker hypothesis FPT = W[1]. Building on the techniques of Araújo et al., we prove that when G is a forest, the decision problem of computing σ(G, w) is W[1]-hard parameterized by the size of a largest connected component of G, and that computing σ(G, w; r) is W[2]-hard parameterized by r. Our results rule out the existence of FPT algorithms for computing these invariants on trees or forests for many natural choices of the parameter.
Fichier principal
Vignette du fichier
TCS-D-17-00413-R1-final.pdf (474.95 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01733595 , version 1 (19-01-2021)

Identifiants

Citer

Julio Araujo, Julien Baste, Ignasi Sau. Ruling out FPT algorithms for Weighted Coloring on forests. Electronic Notes in Discrete Mathematics, 2017, 62, pp.195-200. ⟨10.1016/j.endm.2017.10.034⟩. ⟨hal-01733595⟩
144 Consultations
18 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More