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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 pairof marks are different from each other and the total length of the ruler isminimized. The Golomb ruler problem has applications in information theory,astronomy and communications, and it can be seen as a challenge forcombinatorial optimization algorithms. Although constructing high qualityrulers is well-studied, proving optimality is a far more challenging task. Inthis paper, we provide a computational comparison of different optimizationparadigms, each using a different model (linear integer, constraint programmingand quadratic integer) to certify that a given Golomb ruler is optimal. Wepropose several enhancements to improve the computational performance of eachmethod by exploring bound tightening, valid inequalities, cutting planes andbranching strategies. We conclude that a certain quadratic integer programmingmodel solved through a Benders decomposition and strengthened by two types ofvalid inequalities performs the best in terms of solution time for small-sizedGolomb ruler problem instances. On the other hand, a constraint programmingmodel improved by range reduction and a particular branching strategy couldhave more potential to solve larger size instances due to its promisingparallelization features.

A Computational Comparison of Optimization Methods for the Golomb Ruler Problem