## 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.

Original language | English |
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Title of host publication | Combinatorial Pattern Matching - 23rd Annual Symposium, CPM 2012, Proceedings |

Pages | 427-438 |

Number of pages | 12 |

DOIs | |

State | Published - 2012 |

Event | 23rd Annual Symposium on Combinatorial Pattern Matching, CPM 2012 - Helsinki, Finland Duration: 3 Jul 2012 → 5 Jul 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7354 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 23rd Annual Symposium on Combinatorial Pattern Matching, CPM 2012 |
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Country/Territory | Finland |

City | Helsinki |

Period | 3/07/12 → 5/07/12 |

### 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.

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science