Approximating Minimum Sum Coloring with Bundles

In the Minimum Sum Coloring with Bundles problem, we are given an undirected graph G = (V,E) and (not necessarily disjoint) bundles V_1, V_2, …, V_p ⊆ V with associated weights w_1, …, w_p ≥ 0. The goal is to give a proper coloring of G using positive integers to minimize the weighted average/total completion time of all bundles, where the completion time of a bundle is the maximum integer assigned to one of its nodes. This is a common generalization of the classic Minimum Sum Coloring problem, i.e. when all bundles are singleton nodes, and the classic Chromatic Number problem, i.e. the only bundle is all of V. Despite its generality as an extension of Minimum Sum Coloring, only very special cases have been studied with the most common being the line graph L(H) of a graph H (also known as Coflow Scheduling). We provide the first constant-factor approximation in perfect graphs and, more generally, graphs whose chromatic number is within a constant factor of the maximum clique size in any induced subgraph. For example, we obtain constant-factor approximations for graphs that are well-studied in minimum sum coloring such as interval graphs and unit disk graphs. Next, we extend our results to get constant-factor approximations for a general model where the bundles are disjoint (i.e. can be thought of as jobs brought by the corresponding client) and we are only permitted to color/schedule a bounded number of jobs from each bundle at any given time. Specifically, we get constant-factor approximations for this model if the nodes of graph G have an ordering v_1, v_2, …, v_n such that the left-neighborhood N_𝓁(v_i) : = {v_j : j < i, v_iv_j ∈ E} can be covered by O(1) cliques. For example, this applies to chordal graphs, unit disc graphs, and circular arc graphs.