Minimum Common String Partition Problem: Hardness and Approximations

Abstract

String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing and compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement. A partition of a string $A$ is a sequence ${\cal P} = (P_1,P_2,\dots,P_m)$ of strings, called the blocks, whose concatenation is equal to $A$. Given a partition ${\cal P}$ of a string $A$ and a partition ${\cal Q}$ of a string $B$, we say that the pair $\langle{{\cal P},{\cal Q}}\rangle$ is a common partition of $A$ and $B$ if ${\cal Q}$ is a permutation of ${\cal P}$. The minimum common string partition problem (MCSP) is to find a common partition of two strings $A$ and $B$ with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most $k$ times in each input string, is denoted by $k$-MCSP. In this paper, we show that $2$-MCSP (and therefore MCSP) is NP-hard and, moreover, even APX-hard. We describe a $1.1037$-approximation for $2$-MCSP and a linear time $4$-approximation algorithm for $3$-MCSP. We are not aware of any better approximations.

Approximating Weighted Duo-Preservation in Comparative Genomics

“…Our Result. Notice that the MWDSM problem is NP-hard as its unweighted variant (i.e., the MDSM problem) is known to be NP-hard [14]. We note that the previous approximation algorithms for the MDSM problem do not apply to the MWDSM problem.…”

“…In the simplest form, the only edit operation that is allowed in computing the edit distance is to shift a block of characters; that is, to change the order of the characters in the string. Computing the edit distance under this operation reduces to the Minimum Common String Partition (MCSP) problem, which was introduced by Goldstein et al [14] (see also [21]) and is defined as follows. For a string s, let P (s) denote a partition of s. Given two strings s 1 and s 2 each of length n, where s 2 is a permutation of s 1 , the objective of the MCSP problem is to find a partition P (s 1 ) of s 1 and P (s 2 ) of s 2 of minimum cardinality such that P (s 2 ) is a permutation of P (s 1 ).…”

“…For a string s, let P (s) denote a partition of s. Given two strings s 1 and s 2 each of length n, where s 2 is a permutation of s 1 , the objective of the MCSP problem is to find a partition P (s 1 ) of s 1 and P (s 2 ) of s 2 of minimum cardinality such that P (s 2 ) is a permutation of P (s 1 ). The problem is known to be NP-hard and even APX-hard [14].…”

“…Related Work. Since the MCSP problem is NP-hard [14], there has been many works on designing polynomial-time approximation algorithms for this problem [10,11,12,14,17]. The best approximation results thus far are an O(log n log * n)-approximation algorithm for the general version of the problem [12], and an O(k)-approximation for the k-MCSP problem [12] (the k-MCSP is a variant of the problem in which each character can appear at most k times in each string).…”

“…In terms of the parameterized complexity, the problem is known to be fixed-parameter tractable with respect to k and the size of an optimal partition [6,16], as well as the size of an optimal solution only [7]. For the MDSM problem, we observe that since the MCSP problem is NP-hard [14], the MDSM problem (i.e., its maximization version) is also NP-hard, and in fact even APX-hard [4]. Moreover, the problem is also shown to be fixed-parameter tractable with respect to the number of duos preserved [2].…”