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 constraint-based 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.