Clique Clustering Yields a PTAS for max-Coloring Interval Graphs

We are given an interval graph G = (V,E) where each interval I ∈ V has a weight wI ∈ R+. The goal is to color the intervals V with an arbitrary number of color classes C1,C2,..., Ck such that Σi=1k maxI∈Ci wI 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+e)-approximation algorithm for any e > 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.

[1]  Wolfgang Maass,et al.  Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[2]  Leah Epstein,et al.  On the max coloring problem , 2012, Theor. Comput. Sci..

[3]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[4]  Vangelis Th. Paschos,et al.  Weighted coloring on planar, bipartite and split graphs: Complexity and approximation , 2009, Discret. Appl. Math..

[5]  Xuding Zhu,et al.  A Coloring Problem for Weighted Graphs , 1997, Inf. Process. Lett..

[6]  Vangelis Th. Paschos,et al.  Weighted Coloring: Further Complexity and Approximability Results , 2005, ICTCS.

[7]  Gerd Finke,et al.  Batch processing with interval graph compatibilities between tasks , 2005, Discret. Appl. Math..

[8]  Evripidis Bampis,et al.  Bounded Max-colorings of Graphs , 2010, ISAAC.

[9]  Rajiv Raman,et al.  Max-coloring and online coloring with bandwidths on interval graphs , 2011, TALG.

[10]  Hadas Shachnai,et al.  Batch Coloring Flat Graphs and Thin , 2008, SWAT.

[11]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[12]  Tim Nonner Capacitated max -Batching with Interval Graph Compatibilities , 2010, SWAT.

[13]  Rajiv Raman,et al.  Approximation Algorithms for the Max-coloring Problem , 2005, ICALP.

[14]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[15]  Uriel Feige,et al.  Zero knowledge and the chromatic number , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[16]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[17]  Telikepalli Kavitha,et al.  Max-coloring paths: tight bounds and extensions , 2012, J. Comb. Optim..

[18]  Rajiv Raman,et al.  Buffer minimization using max-coloring , 2004, SODA '04.