MPI-I-96-1-019

This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios $2 {2\over 3}$ ($\approx 2.67$) and $2 {25\over 42}$ ($\approx 2.596$), improving the best previously published $2 {3\over 4}$ approximation. The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings, and then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semi-infinite string $\alpha = a_1 a_2 \cdots$ of period $q$, there exists an integer $k$, such that for {\em any} (finite) string $s$ of period $p$ which is {\em inequivalent} to $\alpha$, the overlap between $s$ and the {\em rotation} $\alpha[k] = a_k a_{k+1} \cdots$ is at most $p+{1\over 2}q$. Moreover, if $p \leq q$, then the overlap between $s$ and $\alpha[k]$ is not larger than ${2\over 3}(p+q)$. In the previous shortest superstring algorithms $p+q$ was used as the standard bound on overlap between two strings with periods $p$ and $q$.