Seed Optimization Is No Easier than Optimal Golomb Ruler Design

Spaced seed is a filter method invented to efficiently identify the regions of interest in similarity searches. It is now well known that certain spaced seeds hit (detect) a randomly sampled similarity region with higher probabilities than the others. Assume each position of the similarity region is identity with probability p independently. The seed optimization problem seeks for the optimal seed achieving the highest hit probability with given length and weight. Despite that the problem was previously shown not to be NP-hard, in practice it seems difficult to solve. The only algorithm known to compute the optimal seed is still exhaustive search in exponential time. In this article we put some insight into the hardness of the seed design problem by demonstrating the relation between the seed optimization problem and the optimal Golomb ruler design problem, which is a well known difficult problem in combinatorial design.