Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Minimum Common String Partition Problem: Hardness and Approximations. | Minimum Common String Partition Problem Hardness and Approximations Avraham Goldstein Department of Mathematics Stony Brook University Stony Brook NY . avUgoldstein@ Petr Kolman Department of Applied Mathematics Faculty of Mathematics and Physics Charles University Prague Czech Republic kolman@ Jie Zhengi Department of Computer Science University of California Riverside CA . zjie@ Submitted Aug 19 2004 Accepted Aug 28 2005 Published Sep 29 2005 Mathematics Subject Classifications 68W25 68Q25 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 P Pl P2 . Pm 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 Qi 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 for 2-MCSP and a linear time 4-approximation algorithm for 3-MCSP. We are not aware of any better approximations. Preliminary version of this work was presented at the 15th International Symposium on Algorithms and Computation 9 . Research done while visiting University of California at Riverside. Partially supported by project 1M0021620808 of MSMT CR and NSF grants CCR-0208856 and ACI-0085910. 1 Supported by NSF grant .
