Algorithmic Aspects of Golomb Ruler Construction

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.