Bounded Max-Colorings of Graphs

TL;DR

This paper studies bounded max-coloring problems in graphs, providing complexity results and approximation algorithms for various graph classes, with specific bounds and optimality results for trees, bipartite graphs, and general graphs.

Contribution

It introduces new approximation algorithms and complexity results for bounded max-coloring problems, including NP-completeness for trees and tight bounds for bipartite graphs.

Findings

Polynomial algorithms for trees with fixed colors

Approximation algorithms with specific ratios for bipartite graphs

NP-completeness of max-edge-coloring on trees

Abstract

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most $b$ and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes' weights. In this paper we present complexity results and approximation algorithms for those problems on general graphs, bipartite graphs and trees. We first show that both problems are polynomial for trees, when the number of colors is fixed, and $H_b$ approximable for general graphs, when the bound $b$ is fixed. For the bounded max-vertex-coloring problem, we show a 17/11-approximation algorithm for bipartite graphs, a PTAS for trees as well as for bipartite graphs when $b$ is fixed. For unit weights, we show that the known 4/3 lower bound for bipartite graphs is tight by providing a simple 4/3 approximation algorithm. For the bounded max-edge-coloring problem, we prove approximation factors of $3-2/\sqrt{2b}$, for general graphs, $\min\{e, 3-2/\sqrt{b}\}$, for bipartite graphs, and 2, for trees. Furthermore, we show that this problem is NP-complete even for trees. This is the first complexity result for max-coloring problems on trees.