Complexities of the Centre and Median String Problems

Abstract

Abstract. Given a finite set of strings, the median string problem consists in finding a string that minimizes the sum of the distances to the strings in the set. Approximations of the median string are used in a very broad range of applications where one needs a representative string that summarizes common information to the strings of the set. It is the case in Classification, in Speech and Pattern Recognition, and in Computational Biology. In the latter, median string is related to the key problem of Multiple Alignment. In the recent literature, one finds a theorem stating the NP-completeness of the median string for unbounded alphabets. However, in the above mentioned areas, the alphabet is often finite. Thus, it remains a crucial question whether the median string problem is NP-complete for finite and even binary alphabets. In this work, we provide an answer to this question and also give the complexity of the related centre string problem. Moreover, we study the parametrized complexity of both problems with respect to the number of input strings.

Finding the Jaccard Median

“…The hardness of finding the best median for a set of points out of a (typically exponentially) large set of candidates strictly depends on the metric in consideration. For instance, the median problem has been shown to be hard for edit distance on strings [8,19], for the Kendall metric on permutations [2,9], but can be solved in polynomial time for the Hamming distance on sets (and more generally, for the ℓ 1 distance on real vectors), and for the Spearman footrule metric on permutations [9]. The general metric -median problem has also been studied in the literature; see, for example, [1,6].…”

Hardness results for the center and median string problems under the weighted and unweighted edit distances

“…In Section 2.1 (see also [21]), we have shown that CENTER STRING under Levenshtein distance is NP-complete and W[1]-hard with respect to the number of input strings, even for binary alphabet. In Section 3, we generalize these results to any weighted edit distance that satisfies a natural condition.…”

“…These results already appear in [21]. We start with the less technical of the two proofs, the one concerning CENTER STRING (Theorem 3).…”

Clustering Methods Based on Closest String via Rank Distance

“…hydrophoby (for instance, this is the case in some protocols that identify similar DNA sequences [21]). In [22,23] it is shown that closest string and median string are NP-hard for finite and even binary alphabets. The existence of fast exact algorithms, when the number of input strings is fixed, is investigated in [22].…”