Hardness of Optimal Spaced Seed Design

Abstract

Speeding up approximate pattern matching is a line of research in stringology since the 80's. Practically fast approaches belong to the class of filtration algorithms, in which text regions dissimilar to the pattern are first excluded, and the remaining regions are then compared to the pattern by dynamic programming. Among the conditions used to test similarity between the regions and the pattern, many require a minimum number of common substrings between them. When only substitutions are taken into account for measuring dissimilarity, counting spaced subwords instead of substrings improves the filtration efficiency. However, a preprocessing step is required to design one or more patterns, called spaced seeds (or gapped seeds), for the subwords, depending on the search parameters. Two distinct lines of research appear the literature: one with probabilistic formulations of seed design problems, in which one wishes for instance to compute a seed with the highest probability to detect the desired similarities (lossy filtration), a second line with combinatorial formulations, where the goal is to find a seed that detects all or a maximum number of similarities (both lossless and lossy filtration). We concentrate on combinatorial seed design problems and consider formulations in which the set of sought similarities is either listed explicitly (RSOS), or characterised by their length and maximal number of mismatches (Non Detection). Several articles exhibit exponential algorithms for these problems. In this work, we provide hardness and inapproximability results for several seed design problems, thereby justifying the complexity of known algorithms. Moreover, we introduce a new formulation of seed design (MWLS), in which the weight of the seed has to be maximised, and show it is as difficult to approximate as Maximum Independent Set.

mpscan: Fast Localisation of Multiple Reads in Genomes

“…For instance, a set of 4 spaced seeds of weight 13 was manually designed to search for 33 bp tags [20]. Hence, adaptation of ZOOM to a new setup re-quires the design of specific seeds, which is a theoretically hard and practically difficult problem [25,26,27]. The present limitation of ZOOM to patterns up to 64 bp is certainly due to this bottleneck.…”

Exact Search Algorithms for Biological Sequences

“…Hence, sets of spaced seeds are specifically designed for a certain read/match length and a maximum number of allowed differences, and different sets corresponding to different parameter combinations are hard coded in ZOOM. All known formulations of the seed design problem are at least NP-hard, even for a single seed [35,42,47].…”

Rapid Homology Search with Neighbor Seeds

“…For a given spaced seed, computing the probability that it generates a hit in a homology region is NP-hard in the Bernoulli model [28], as well as in the uniform model [29].…”

“…Then the relaxed seed is equivalent to the spaced seed S ∪ S . Computing the sensitivity is NP-hard in both models [28,29].…”