On the Size of Systems of Sets Every <i>t</i> of which Have an SDR, with an Application to the Worst-Case Ratio of Heuristics for Packing Problems

Hurkens, Caj Cor; Schrijver, Alexander

doi:10.1137/0402008

Cited by 236 publications

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“…Furthermore, for some positive constant ε > 0, finding an approximation of the size of maximum independent set within a factor of n ε is NP-hard [1]. For some classes of graphs, there exist approximation algorithms with a constant factor [3], [9], [11], [14].…”

Section: Approximationmentioning

confidence: 99%

Transforming Graphs with the Same Graphic Sequence

Bereg

Ito

2017

Journal of Information Processing

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Section: Approximationmentioning

confidence: 99%

Transforming Graphs with the Same Graphic Sequence

Bereg

Ito

2017

Journal of Information Processing

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“…Using a similar argument as in Lemma 3.1, the MINBUSY problem is equivalent to g-dimensional matching, which admits a (g/2 + )-approximation [14]. By Lemma 2.1 this implies a (g−2+2/g)-approximation for MINBUSY.…”

Section: A Clique Instancesmentioning

confidence: 83%

Optimizing busy time on parallel machines

Mertzios

Shalom

Voloshin

et al. 2015

Theoretical Computer Science

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Abstract-We consider the following fundamental scheduling problem in which the input consists of n jobs to be scheduled on a set of identical machines of bounded capacity g (which is the maximal number of jobs that can be processed simultaneously by a single machine). Each job is associated with a start time and a completion time; it is supposed to be processed from the start time to the completion time (and in one of our extensions it has to be scheduled also in a continuous number of days; this corresponds to a two-dimensional version of the problem). We consider two versions of the problem. In the scheduling minimization version the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version the goal is to maximize the number of jobs that are scheduled for processing under a budget constraint given in terms of busy time. This is the first study of the maximization version of the problem. The minimization problem is known to be NP-Hard, thus the maximization problem is also NP-Hard. We consider various special cases, identify cases where an optimal solution can be computed in polynomial time, and mainly provide constant factor approximation algorithms for both minimization and maximization problems. Some of our results improve upon the best known results for this job scheduling problem. Our study has applications in power consumption, cloud computing and optimizing switching cost of optical networks.

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“…For k = 4, this is not the case. The approximation ratio of the Restricted 4-Set Packing algorithm is 7 16 , which is worse than the 1 2 − ǫ ratio of the 4-set packing heuristic but it is also tight. For k = 3, we use the semi-local optimization technique [2].…”

Section: Introductionmentioning

confidence: 97%

“…For k ≥ 7, we use the k-set packing heuristic introduced by Hurkens and Shrijver [7], which achieves the best known to date approximation ratio 2 k − ǫ for the k-Set Packing problem for any ǫ > 0. On the other hand, the best hardness result by Hazan et al [6] shows that it is NP-hard to approximate the k-Set Packing problem within Ω( ln k k ).…”

Section: Introductionmentioning

confidence: 99%

“…We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted k-Set Packing algorithm, our k-Set Cover algorithm is composed of the k-Set Packing heuristic [7] for k ≥ 7, Restricted k-Set Packing for k = 6, 5, 4 and the semi-local (2, 1)-improvement [2] for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of H k − 0.6402 + Θ( 1 k ), where H k is the k-th harmonic number.…”

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Packing-Based Approximation Algorithm for the k-Set Cover Problem

Fürer

2011

Algorithms and Computation

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Abstract. We present a packing-based approximation algorithm for the k-Set Cover problem. We introduce a new local search-based k-set packing heuristic, and call it Restricted k-Set Packing. We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted k-Set Packing algorithm, our k-Set Cover algorithm is composed of the k-Set Packing heuristic [7] for k ≥ 7, Restricted k-Set Packing for k = 6, 5, 4 and the semi-local (2, 1)-improvement [2] for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of H k − 0.6402 + Θ( 1 k ), where H k is the k-th harmonic number. For small k, our results are 1.8667 for k = 6, 1.7333 for k = 5 and 1.5208 for k = 4. Our algorithm improves the currently best approximation ratio for the k-Set Cover problem of any k ≥ 4.

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On the Size of Systems of Sets Every t of which Have an SDR, with an Application to the Worst-Case Ratio of Heuristics for Packing Problems

Cited by 236 publications

References 0 publications

Transforming Graphs with the Same Graphic Sequence

Transforming Graphs with the Same Graphic Sequence

Optimizing busy time on parallel machines

Packing-Based Approximation Algorithm for the k-Set Cover Problem

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