Clique Clustering Yields a PTAS for max-Coloring Interval Graphs

We are given an interval graph G = (V, E) where each interval I is an element of V has a weight wI is an element of R+. The goal is to color the intervals V with an arbitrary number epsilon of color classes C-1, C-2,...,C-k such that Sigma(k)(i=1) max(I is an element of Ci) w(I) is minimized. This problem, called max-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 (SODA' 04) and presented a 2-approximation algorithm. Closing a gap which has been open for years, 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. Besides using standard preprocessing techniques such as geometric rounding and shifting, our main building block is a general technique for trading the overlap structure of an interval graph for accuracy, which we call clique clustering.