On the complexity of constructing Golomb Rulers

Exploiting a hypergraph model for finding Golomb rulers

“…This has been open for at least nine years. Moreover there are numerous natural variations of Golomb ruler construction [14,19]. In this paper, we focused on GSR introduced by Meyer and Papakonstantinou [14].…”

“…Meyer and Papakonstantinou [14] showed that Golomb Subruler (GSR) is NPhard via a reduction from an NP-hard SAT variant. However, the construction of the ruler corresponding to the SAT formula is involved and hard to comprehend.…”

“…The number of marks on a ruler is called its order n and the distance between its smallest mark (which can be assumed to be 0 without loss of generality) and its largest mark is called its length. While it is easy to construct Golomb rulers, finding a shortest Golomb ruler for a given order n (encoded in unary) is assumed to be computationally intractable [6,14]. However, there is no NP-hardness result for this problem so far, but due to the multitude of its practical applications, there have been several computational studies based on heuristics and massive parallelism (see [7,8,17] for example).…”

“…Meyer and Papakonstantinou [14] formalized this as the Golomb Subruler problem and showed its NP-hardness.…”

“…However, there is no NP-hardness result for this problem so far, but due to the multitude of its practical applications, there have been several computational studies based on heuristics and massive parallelism (see [7,8,17] for example). The applications of Golomb ruler construction include radio frequency selection, radio astronomy, data encryption, communication networks, and bioinformatics [2, 3,4,14,18]. For instance, when placing radio channels in the frequency spectrum, intermodulation interference is introduced by nonlinear behavior in transmitters and receivers.…”