Abstract
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.
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Nonner, T. (2011). Clique Clustering Yields a PTAS for max-Coloring Interval Graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_16
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DOI: https://doi.org/10.1007/978-3-642-22006-7_16
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