Algorithmic Aspects of Golomb Ruler Construction

TL;DR

This paper explores the computational complexity and structural properties of Golomb rulers, providing simplified hardness proofs, hypergraph characterizations, and insights into their construction challenges.

Contribution

It offers a simplified NP-hardness proof, a hypergraph-based structural analysis, and a fixed-parameter approach for Golomb ruler construction problems.

Findings

NP-completeness of optimal Golomb ruler construction

Hypergraph characterization reveals structural properties

Fixed-parameter approach enables problem kernel development

Abstract

We consider Golomb rulers and their construction. Common rulers feature marks at every unit measure, distances can often be measured with numerous pairs of marks. On Golomb rulers, for every distance there are at most two marks measuring it. The construction of optimal---with respect to shortest length for given number of marks or maximum number of marks for given length---is nontrivial, various problems regarding this are NP-complete. We give a simplified hardness proof for one of them. We use a hypergraph characterization of rulers and Golomb rulers to illuminate structural properties. This gives rise to a problem kernel in a fixed-parameter approach to a construction problem. We also take a short look at the practical implications of these considerations.