Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

Abstract

We are given an interval graph $G = (V,E)$ where each interval $I \in V$ has a weight $w_I \in \mathbb {Q}^+$. The goal is to color the intervals $V$ with an arbitrary number of color classes $C_1, C_2, \ldots , C_k$ such that $\sum _{i=1}^k \max _{I \in C_i} w_I$ is minimized. This problem, called max-coloring interval graphs or weighted coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard [21] and presented a 2-approximation algorithm. We settle the approximation complexity of this problem by giving a polynomial-time approximation scheme (PTAS), that is, we show that there is an $(1+\epsilon )$-approximation algorithm for any $\epsilon > 0$. The PTAS also works for the bounded case where the sizes of the color classes are bounded by some arbitrary $k \le n$.