Capacitated max -Batching with Interval Graph Compatibilities

Nonner, Tim

doi:10.1007/978-3-642-13731-0_18

Cited by 1 publication

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References 13 publications

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“…For instance, maxcoloring co-interval graphs is polynomially solvable [8,3], and only the capacitated case described above is NP-hard [15]. However, except for this preprocessing step, all arguments in this paper differ completely from [15]. Bambis et al [2] gave some results for the max-coloring problem with capacities.…”

Section: Introductionmentioning

confidence: 90%

“…Despite the similarity in name, it is worth mentioning here that max-coloring interval graphs and max-coloring co-interval graphs have a quite different structure. For instance, maxcoloring co-interval graphs is polynomially solvable [8,3], and only the capacitated case described above is NP-hard [15]. However, except for this preprocessing step, all arguments in this paper differ completely from [15].…”

Section: Introductionmentioning

confidence: 95%

“…We repeat the proof from [15] in compressed form. First, it is easy to see that we can apply a geometric rounding step such that, for any α > 1, by losing an α-factor in the approximation ratio, we may assume that w i = α i−1 for each level i.…”

Section: Lemma 2 Combine Lemmas 8 Andmentioning

confidence: 99%

“…Lemma 1 has already been applied by the author [15] to obtain a PTAS for the inverse problem of max-coloring co-interval graphs with constant capacities, i.e., there is a constant bound k such that we additionally require that |C| ≤ k for any color class C. Note that finding a max-coloring of a co-interval graph G is equivalent to finding a max-clique partition of the co-graph of G, which is again an interval graph. Despite the similarity in name, it is worth mentioning here that max-coloring interval graphs and max-coloring co-interval graphs have a quite different structure.…”

Section: Introductionmentioning

confidence: 99%

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Clique Clustering Yields a PTAS for max-Coloring Interval Graphs

Nonner

2011

Automata, Languages and Programming

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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 k i=1 max I∈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 + ǫ)-approximation algorithm for any ǫ > 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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