Optimum Solution of the Closest String Problem via Rank Distance

Abstract

The Closest String Problem (CSP) calls for finding an n-string that minimizes its maximum distance from m given n-strings. Integer linear programming (ILP) proved to be able to solve large CSPs under the Hamming distance, whereas for the Levenshtein distance, preferred in computational biology, no ILP formulation has so far be investigated. Recent research has however demonstrated that another metric, rank distance, can provide interesting results with genomic sequences. Moreover, CSP under rank distance can easily be modeled via ILP: optimal solutions can then be certified, or information on approximation obtained via dual gap. In this work we test this ILP formulation on random and biological data. Our experiments, conducted on strings with up to 600 nucleotides, show that the approach outperforms literature heuristics. We also enforce the formulation by cover inequalities. Interestingly, due to the special structure of the rank distance between two strings, cover separation can be done in polynomial time.