String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing or 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 P = (P_1, P_2, ... , P_m) of strings, called the blocks, whose concatenation is equal to A. Given a partition P of a string A and a partition Q of a string B, we say that the pair < P, Q > is a common partition of A and B if Q is a permutation of 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.
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