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Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

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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 \( k \le n \).

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Notes

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    All logarithms have base 2.

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Correspondence to Tim Nonner.

Additional information

A preliminary version of this paper has appeared at ICALP’11.

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Nonner, T. Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs. Algorithmica 80, 2941–2956 (2018). https://doi.org/10.1007/s00453-017-0362-9

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Keywords

  • Combinatorial optimization
  • Approximation algorithms
  • Approximation schemes
  • Graph algorithms
  • Interval graphs