# A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

## TL;DR

This paper compares various optimization methods for solving the Golomb ruler problem, highlighting the most efficient models and strategies for different problem sizes through computational experiments.

## Contribution

It provides a comprehensive computational comparison of linear, constraint programming, and quadratic models, with enhancements to improve solution efficiency and scalability.

## Key findings

- Quadratic integer programming with Benders decomposition is most effective for small instances.
- Constraint programming with range reduction shows promise for larger instances.
- Enhanced models significantly reduce solution times through bound tightening and valid inequalities.

## 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.