A Lagrangian Relaxation for Golomb Rulers

Slusky, Marla R.; Hoeve, Willem‐Jan van

doi:10.1007/978-3-642-38171-3_17

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…sets of integers so that all pairwise sums of elements of the set are distinct. Lower bounds for the GRP and connections to number theory are discussed, respectively, in [14,23] and [7,22].…”

Section: Introductionmentioning

confidence: 99%

A conjecture on a continuous optimization model for the Golomb Ruler Problem

Duxbury

Neto

2021

RAIRO-Oper. Res.

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A Golomb Ruler (GR) is a set of integer marks along an imaginary ruler such that all the distances of the marks are different. Computing a GR of minimum length is associated to many applications (from astronomy to information theory). Although not yet demonstrated to be NP-hard, the problem is computationally very challenging. This brief note proposes a new continuous optimization model for the problem and, based on a given theoretical result and some computational experiments, we conjecture that an optimal solution of this model is also a solution to an associated GR of minimum length.

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Section: Introductionmentioning

confidence: 99%

A conjecture on a continuous optimization model for the Golomb Ruler Problem

Duxbury

Neto

2021

RAIRO-Oper. Res.

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“…In general, constructing a Golomb ruler with a given number of marks is an easy task, and there are many heuristic methods that provide high quality rulers. For instance, previous literature on heuristics has focused on affine and projective plane constructions [15,7], genetic algorithms [19], and local search [13,6] while exact methods based on constraint programming [17,8] or hybrid methods [16] exist as well. Although not proven to be NP-hard yet, solving the Golomb ruler problem exactly proved to be notoriously difficult.…”

Section: Introductionmentioning

confidence: 99%

“…As a comparison, it took 2.8 hours for the constraint programming model in [8] to find an optimal ruler for the 13-mark instance and another 11.8 hours to prove its optimality. The lean implementation of the search method in [16] reduced the respective computational effort to 0.6 and 1.3 hours, albeit at the expense of a significantly larger search tree.…”

Section: Introductionmentioning

confidence: 99%

A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

Kocuk

Hoeve

2019

Integration of Constraint Programming, Artificial Intelligence, and Operations Research

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The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is minimized. The Golomb ruler problem has applications in information theory, astronomy and communications, and it can be seen as a challenge for combinatorial optimization algorithms. Although constructing high quality rulers is well-studied, proving optimality is a far more challenging task. In this paper, we provide a computational comparison of different optimization paradigms, each using a different model (linear integer, constraint programming and quadratic integer) to certify that a given Golomb ruler is optimal. We propose several enhancements to improve the computational performance of each method by exploring bound tightening, valid inequalities, cutting planes and branching strategies. We conclude that a certain quadratic integer programming model solved through a Benders decomposition and strengthened by two types of valid inequalities performs the best in terms of solution time for small-sized Golomb ruler problem instances. On the other hand, a constraint programming model improved by range reduction and a particular branching strategy could have more potential to solve larger size instances due to its promising parallelization features.

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“…1 The bounds for the new propagator are based on a non-linear Lagrangian relaxation, with the multipliers being optimized via a subgradient method. Lagrangian relaxation has proved to be a powerful technique for performing reduced-cost filtering in CP, mainly in the context of linear relaxations of integer programs (e.g., [32][33][34]). The non-linear Lagrangian relaxation used in this paper is inspired by the work presented in [19], for solving resource constrained shortest path problems with a super additive objective function.…”

Section: Introductionmentioning

confidence: 99%

A lagrangian propagator for artificial neural networks in constraint programming

Lombardi

Gualandi²

2015

Constraints

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This paper discusses a new method to perform propagation over a (two-layer, feed-forward) Neural Network embedded in a Constraint Programming model. The method is meant to be employed in Empirical Model Learning, a technique designed to enable optimal decision making over systems that cannot be modeled via conventional declarative means. The key step in Empirical Model Learning is to embed a Machine Learning model into a combinatorial model. It has been showed that Neural Networks can be embedded in a Constraint Programming model by simply encoding each neuron as a global constraint, which is then propagated individually. Unfortunately, this decomposition approach may lead to weak bounds. To overcome such limitation, we propose a new network-level propagator based on a non-linear Lagrangian relaxation that is solved with a subgradient algorithm. The method proved capable of dramatically reducing the search tree size on a thermal-aware dispatching problem on multicore CPUs. The overhead for optimizing the Lagrangian multipliers is kept within a reasonable level via a few simple techniques. This paper is an extended version of [27], featuring an improved structure, a new filtering technique for the network inputs, a set of overhead reduction techniques, and a thorough experimentation.

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A Lagrangian Relaxation for Golomb Rulers

Cited by 5 publications

References 25 publications

A conjecture on a continuous optimization model for the Golomb Ruler Problem

A conjecture on a continuous optimization model for the Golomb Ruler Problem

A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

A lagrangian propagator for artificial neural networks in constraint programming

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