A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

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

A Lagrangian Relaxation for Golomb Rulers

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.</p

A Lagrangian Relaxation for Golomb Rulers

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.