Shorter Path Constraints for the Resource Constrained Shortest Path Problem

Gellermann, Thorsten; Sellmann, Meinolf; Wright, Robert

doi:10.1007/11493853_16

Cited by 11 publications

(3 citation statements)

References 13 publications

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“…We note that Lagrangian relaxations have been applied before in the context of CP, see for example [16,17,18,19,20,21,22,23]. Our results further strengthen the idea that Lagrangian relaxations are a particularly useful method from operations research for enhancing the inference process of constraint programming.…”

Section: Introductionsupporting

confidence: 75%

A Lagrangian Relaxation for Golomb Rulers

Slusky

Hoeve

2013

Lecture Notes in Computer Science

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Abstract. The Golomb Ruler Problem asks to position n integer marks on a ruler such that all pairwise distances between the marks are distinct and the ruler has minimum total length. It is a very challenging combinatorial problem, and provably optimal rulers are only known for n up to 26. Lower bounds can be obtained using Linear Programming formulations, but these are computationally expensive for large n. In this paper, we propose a new method for finding lower bounds based on a Lagrangian relaxation. We present a combinatorial algorithm that finds good bounds quickly without the use of a Linear Programming solver. This allows us to embed our algorithm into a constraint programming search procedure. We compare our relaxation with other lower bounds from the literature, both formally and experimentally. We also show that our relaxation can reduce the constraint programming search tree considerably.

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Section: Introductionsupporting

confidence: 75%

A Lagrangian Relaxation for Golomb Rulers

Slusky

Hoeve

2013

Lecture Notes in Computer Science

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“…Since the problem is NP-hard in the general case, we introduce shortest-path relaxations and develop and compare different filtering algorithms for different graph classes. This paper is based on our findings in [20,34]. In Section 2, we review the notion of relaxed consistency, and in Section 3, we define shorter path constraints formally.…”

Section: Introductionmentioning

confidence: 99%

Cost-based Filtering for Shorter Path Constraints

2006

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Many real world problems, e.g. personnel scheduling and transportation planning, can be modeled naturally as Constrained Shortest Path Problems (CSPPs), i.e., as Shortest Path Problems with additional constraints. A well studied problem in this class is the Resource Constrained Shortest Path Problem. Reduction techniques are vital ingredients of solvers for the CSPP, that is frequently NP-hard, depending on the nature of the additional constraints. Viewed as heuristics, these techniques have not been studied theoretically with respect to their efficiency, i.e., with respect to the relation of filtering power and running time. Using the concepts of Constraint Programming, we provide a theoretical study of cost-based filtering for shorter path constraints on acyclic, on undirected, and on directed graphs that do not contain negative cycles. We then show empirically how reasoning about path-substructures in combination with CP-based Lagrangian relaxation can help to improve significantly over previously developed problem-tailored filtering algorithms for the resource constrained shortest path problem and investigate the impact of required-edge detection, undirected versus directed filtering, and the choice of the algorithm optimizing the Lagrangian dual.

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“…RCSP problems have been addressed by many authors including Beasley and Christofides (1989), Dumitrescu and Boland (2001), Hassin (1992), and Mehlhorn and Ziegelmann (2000). In particular, RCSP problems of limited size can be solved through special-purpose algorithms such as those developed in Carlyle and Wood (2003) and Gellermann et al (2005), or through efficient generalpurpose algorithms available in commercial optimization software packages.…”

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confidence: 99%

Network Optimization Models for Resource Allocation in Developing Military Countermeasures

Golany

Kress

Penn

et al. 2012

Operations Research

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A military arms race is characterized by an iterative development of measures and countermeasures. An attacker attempts to introduce new weapons in order to gain some advantage, whereas a defender attempts to develop countermeasures that can mitigate or even eliminate the effects of the weapons. This paper addresses the defender's decision problem: given limited resources, which countermeasures should be developed and how much should be invested in their development to minimize the damage caused by the attacker's weapons over a certain time horizon. We formulate several optimization models, corresponding to different operational settings, as constrained shortest-path problems and variants thereof. We then demonstrate the potential applicability and robustness of this approach with respect to various scenarios.

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Shorter Path Constraints for the Resource Constrained Shortest Path Problem

Cited by 11 publications

References 13 publications

A Lagrangian Relaxation for Golomb Rulers

A Lagrangian Relaxation for Golomb Rulers

Cost-based Filtering for Shorter Path Constraints

Network Optimization Models for Resource Allocation in Developing Military Countermeasures

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