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{\bf Avraham Goldstein, Petr Kolman and Jie Zheng}
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{\bf Minimum Common String Partition Problem: Hardness and Approximations}
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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 {\em partition} of a string $A$ is a sequence ${\cal P} =
(P_1,P_2,\dots,P_m)$ of strings, called the \emph{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 {\em common partition}
of $A$ and $B$ if ${\cal Q}$ is a permutation of ${\cal P}$.  The {\em
minimum common string partition} problem ({\tt MCSP}) is to find a
common partition of two strings $A$ and $B$ with the minimum number of
blocks.  The restricted version of {\tt MCSP} where each letter occurs
at most $k$ times in each input string, is denoted by $k$-{\tt MCSP}.

In this paper, we show that $2$-{\tt MCSP} (and therefore {\tt MCSP})
is NP-hard and, moreover, even APX-hard.
We describe a $1.1037$-approximation for $2$-{\tt MCSP}
and a linear time $4$-approximation algorithm for $3$-{\tt MCSP}.
We are not aware of any better approximations.

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