Improved Approximation Results on the Shortest Common Supersequence Problem

  • Authors:

  • Zvi Gotthilf;Moshe Lewenstein

  • Affiliations:

  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel 52900;Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel 52900

  • Venue:
  • SPIRE '09 Proceedings of the 16th International Symposium on String Processing and Information Retrieval
  • Year:
  • 2009

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Abstract

The problem of finding the Shortest Common Supersequence (SCS ) of an arbitrary number of input strings is a well-studied problem. Given a set L of k strings, s 1 , s 2 , ..., s k , over an alphabet Σ, we say that their SCS is the shortest string that contains each of the input strings as a subsequence. The problem is known to be NP-hard [8] even over binary alphabet [12]. In this paper we focus on approximating two NP-hard variants of the SCS problem. For the first variant, where all input strings are of length 2, we present a $2 - \frac {2}{1 + \log{n}\log{\log{n}}}$ approximation algorithm, where |Σ| = n . This result immediately improves the $2 - \frac {4}{n+1}$ approximation algorithm presented in [17]. Moreover, we present a $\frac{7}{6}$ ($\approx 1.166\bar{6}$) approximation algorithm for the restricted variant (but still NP-hard ) where all input strings are of length 2 and every character in Σ has at most 3 occurrences in L .