A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

Abstract

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 conjecture on a continuous optimization model for the Golomb Ruler Problem

“…For n = 5, ..., 28, the values for L u were obtained using the greedy heuristic as described in the previous section. Table 1 gives such values, comparing to two conjectured upper bounds for the GRP [8].…”

“…In order to present the first model to the GRP (first described in [8]), we need an upper bound L u for the length of the ruler, given also as input of the problem, in addition to n marks on the ruler:…”

“…In the literature, there are many heuristics for calculating upper bounds for the GRP (for example, see [8]). We propose a simple greedy heuristic whose pseudocode is given below 1 :…”

“…In [13], a linear programming model is proposed for the GRP and, more recently, authors in [8] compare different optimization approaches to certify optimality to the GRP.…”

The Proof of a Conjecture For a Continuos Golumb Ruler Model

“…More explicitly , we say that a Golumb Ruler problem is to find the smallest length of the ruler such that the n marked points form a Golumb ruler . In [9] , a discrete model for the Golumb ruler problem was firstly presented . Given an positive inerger n and a upper bound L n for the length of the ruler .…”