Clique Clustering Yields a PTAS for max-Coloring Interval Graphs

  • Tim Nonner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


We are given an interval graph G = (V,E) where each interval I ∈ V has a weight w I  ∈ ℝ +  . The goal is to color the intervals V with an arbitrary number of color classes C 1, C 2, …, C k such that \( \sum_{i=1}^k \max_{I \in C_i} w_I \) 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 + ε) -approximation algorithm for any ε > 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.


Polynomial Time Interval Graph Color Class Graph Class Perfect Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tim Nonner
    • 1
  1. 1.IBM ResearchZurichSwitzerland

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