Constraint-based search for optimal Golomb rulers

Abstract

Finding optimal Golomb rulers is an extremely challenging combinatorial problem. The distance between each pair of mark is unique in a Golomb ruler. For a given number of marks, an optimal Golomb ruler has the minimum length. Golomb rulers are used in application areas such as x-ray crystallography, radio astronomy, information theory, and pulse phase modulation. The most recent optimal Golomb ruler search algorithm hybridises a range of techniques such as greedy randomised adaptive search, scatter search, tabu search, clustering techniques, and constraint programming, and obtains optimal Golomb rulers of up to 16 marks with very low success rates. In this paper, we provide tight upper bounds for Golomb ruler marks and present heuristic-based effective domain reduction techniques. Using these along with tabu and configuration checking meta-heuristics, we then develop a constraintbased multi-point local search algorithm to perform a satisfaction search for optimal Golomb rulers of specified length. We then present an algorithm to perform an optimisation search that minimises the length using the satisfaction search repeatedly. Our satisfaction search finds optimal Golomb rulers of up to 19 marks while the optimisation search finds up to 17 marks.

Evaluating logic gate constraints in local search for structured satisfiability problems

“…When symmetries are absent or broken, the effects could be extremely strong, slowing down local search performance by several orders of magnitude. One great example is Golomb Rulers (Prestwich 2002;Polash et al 2017) that have only a handful number of solutions in the enormous search space and local search struggles hard. Because of lack of symmetries, in atype gates, changing the output from the undesired one to the desired one could often need changing more than one input and essentially more than one involving variable.…”

Constraint based local search for flowshops with sequence-dependent setup times

“…To that end, we have employed a constraint based local search approach in our proposed algorithm. Constraint based local search approach is a concept that recently attracts great attention for solving different combinatorial problems such as Golomb rulers (Polash et al, 2017a), all-interval series (Polash et al, 2017b), minimum spanning tree problem (Arbelaez et al, 2018), aircraft scheduling (Riahi et al, 2018a(Riahi et al, , 2019b, and customer order scheduling (Riahi et al, 2019a). Constraint based approaches transform the constraints into an objective function and use search algorithms to optimise the objective (Hentenryck and Michel, 2009).…”