On approximating string selection problems with outliers

  • Authors:

  • Christina Boucher;Gad M. Landau;Avivit Levy;David Pritchard;Oren Weimann

  • Affiliations:

  • Department of Computer Science, University of California, San Diego;Department of Computer Science, University of Haifa, Haifa, Israel,Polytechnic Institute of NYU, Brooklyn, NY;Shenkar College for Engineering and Design, Ramat-Gan, Israel,CRI, University of Haifa, Haifa, Israel;CEMC, University of Waterloo, Canada;Department of Computer Science, University of Haifa, Haifa, Israel

  • Venue:
  • CPM'12 Proceedings of the 23rd Annual conference on Combinatorial Pattern Matching
  • Year:
  • 2012

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Abstract

Many problems in bioinformatics are about finding strings that approximately represent a collection of given strings. We look at more general problems where some input strings can be classified as outliers. The Close to Most Strings problem is, given a set S of same-length strings, and a parameter d, find a string x that maximizes the number of "non-outliers" within Hamming distance d of x. We prove that this problem has no polynomial-time approximation scheme (PTAS) unless NP has randomized polynomial-time algorithms, correcting a decade-old mistake. The Most Strings with Few Bad Columns problem is to find a maximum-size subset of input strings so that the number of non-identical positions is at most k; we show it has no PTAS unless P=NP. We also observe Closest to k Strings has no efficient PTAS (EPTAS) unless the parameterized complexity hierarchy collapses. In sum, outliers help model problems associated with using biological data, but we show the problem of finding an approximate solution is computationally difficult.