006: Golomb rulers

Proposed by Peter van Beek

These problems are said to have many practical applications including sensor placements for x-ray crystallography and radio astronomy. A Golomb ruler may be defined as a set of $m$ integers $0 = a_1 < a_2 < … < a_m$ such that the $m(m-1)/2$ differences $a_j - a_i, 1 <= i < j <= m$ are distinct. Such a ruler is said to contain m marks and is of length $a_m$. The objective is to find optimal (minimum length) or near optimal rulers. Note that a symmetry can be removed by adding the constraint that $a_2 - a_1 < a_m - a_{m-1}$, the first difference is less than the last.

There exist several interesting generalizations of the problem which have received attention like modular Golomb rulers (differences are all distinct mod a given base), disjoint Golomb rulers, Golomb rectangles (the 2-dimensional generalization of Golomb rulers), and difference triangle sets (sets of rulers with no common difference).