Algorithmic Aspects of Golomb Ruler Construction

Golomb rulers have various applications ranging from radio astronomy to cryptography. This explains the interest in computing Golomb rulers that are particularly short for a given number of marks or have many marks when given a maximum length. Unfortunately, from a computational complexity point of view, some of the corresponding decision problems have been proven to be NP-complete, while little is known about other very natural problems.

Despite this, much effort has been made to compute short or dense Golomb rulers and to prove them optimal. Various implementations of exhaustive searches have been given and discussed as well as heuristic and evolutionary approaches. A sophisticated project searches for Golomb rulers through a distributed computer network, enabling users to donate idle computing time.

In this work, we give a short insight into the work that has been done in the field and we briefly consider two natural problems of unsettled computational complexity. We give a natural hypergraph characterization for rulers such that only Golomb rulers correspond to a specific subset of the graphs. We then consider a construction problem that has been proven to be NP-complete. We give a simplified proof for this and then look at two natural parameterizations. For one of the parameterizations, we provide a fixed-parameter algorithm, and some heuristic improvements along with a cubic-size problem kernel that mainly follows from some structure that we observe in characteristic hypergraphs. Finally, we implemented an algorithm that uses the fixed-parameter approach and comment on our experimental results.

According to Colannino [8] and Dimitromanolakis [12], W. C. Babcock first discovered Golomb rulers while analyzing positioning of radio channels in the frequency spectrum. He investigated inter-modulation distortion appearing in For every distance in Golomb rulers, there is at most one pair of marks that measure this distance. For example, the distance one is only measured by the marks 0 and 1 on the Golomb ruler whereas this distance is measured by six pairs of marks on the common ruler. Both rulers measure every integer distance up to their length. Such rulers are called perfect. This, however, is a rare trait among Golomb rulers, as one can proof [12].

consecutive radio bands [3] and observed that when positioning each pair of channels at a distinct distance, then third order distortion was eliminated and fifth order distortion was lessened.

Rankin [29] lists other interesting applications, two of which we touch shortly here. In radio astronomy, arrays of radio telescopes are used to gather information about celestial bodies via interferometry. The telescopes are arranged in a single line, and information is extracted from difference measurements between two telescopes [6]. By placing them at the marks of a Golomb ruler, the number of these measurements and thus information gathered is maximized. This is a special case of a linear array. Linear arrays are also used in other related fields such as antennae construction.

In computer communication networks Golomb rulers can be used to simplify the message passing process. When allocating the node names corresponding to marks on a Golomb ruler, messages do not need to specify both origin and destination addresses. Since the differences between marks in a Golomb ruler are unique, the difference and the direction of arrival suffice to identify the origin and destination node [5].

We now gather a common knowledge base for our considerations. At first we will introduce rulers and Golomb rulers, then go on to hypergraphs, some basic fixed parameter techniques and finally define some notation. We assume the reader to be familiar with basic mathematics and classic complexity theory. There are many recommendable books on complexity theory, see for example [2,27].

Definition 1.2.1 (Ruler). A ruler is a set R := {m i : 1 ≤ i ≤ n} ⊂ Z with m i < m i+1 . The mark m i is called the i’th mark on R. The ruler is said to have n marks and |m n -m 1 | is called the length of the ruler. We call a set R ⊆ R a subruler of R. Definition 1.2.2 (Golomb ruler). A ruler R = {m i : 1 ≤ i ≤ n} is called Golomb ruler if for every d ∈ N \ {0} there is at most one solution to the equation d = m i -m j , m i , m j ∈ R.

It is easy to see that if a ruler {m i : 1 ≤ i ≤ n} is Golomb, so are the rulers {m i c : 1 ≤ i ≤ n} and {m i + c : 1 ≤ i ≤ n} for a constant c ∈ Z.

So for every Golomb ruler R = {m i : 1 ≤ i ≤ n}, there is a Golomb ruler R with only positive marks, starti

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