The min-max edge q-coloring problem

TL;DR

This paper introduces the min-max edge q-coloring problem inspired by wireless networks, proving its NP-hardness, providing exact solutions for specific graph classes, and proposing approximation algorithms for planar graphs.

Contribution

It formally defines the min-max edge q-coloring problem, proves NP-hardness, and offers exact and approximation algorithms for various graph types.

Findings

NP-hardness for q ≥ 2

Polynomial-time exact algorithm for trees

Approximation algorithm for planar graphs

Abstract

In this paper we introduce and study a new problem named \emph{min-max edge $q$-coloring} which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer $q$. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most $q$ different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results: 1. Min-max edge $q$-coloring is NP-hard, for any $q \ge 2$. 2. A polynomial time exact algorithm for min-max edge $q$-coloring on trees. 3. Exact formulas of the optimal solution for cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial lower bound of the optimal solution with respect to the average degree of the graph. 5. An approximation algorithm for planar graphs.