## 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 the 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 erroneous proof made previously in the literature. 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 a 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.

Original language | English |
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Pages (from-to) | 107-114 |

Number of pages | 8 |

Journal | Theoretical Computer Science |

Volume | 498 |

DOIs | |

State | Published - 5 Aug 2013 |

### Bibliographical note

Funding Information:The authors would like to thank Dr. Bin Ma for mentioning the error in his inapproximability proof and encouraging us to work on a correction. We thank Dr. Daniel Lokshtanov, Christine Lo, and the referees for their insights and comments. This work was supported by Natural Sciences and Engineering Research Council of Canada Post Doctoral Fellowship program, the Gerald Schwartz and Heather Reisman Foundation, the Israel Science Foundation grant (347/09), the National Science Foundation Award (0904246), and Grant Number 2008217 from the United States–Israel Binational Science Foundation (BSF) and DFG.

## Keywords

- String algorithms
- String selection

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)