### Abstract

The shortest common superstring and the shortest common supersequence are two well studied problems having a wide range of applications. In this paper we consider both problems with resource constraints, denoted as the Restricted Common Superstring (shortly RCSstr) problem and the Restricted Common Supersequence (shortly RCSseq). In the RCSstr (RCSseq) problem we are given a set S of n strings, s _{1}, s _{2}, ..., s _{n} , and a multiset t = {t _{1}, t _{2}, ..., t _{m} }, and the goal is to find a permutation π: {1, ..., m} → {1, ..., m} to maximize the number of strings in S that are substrings (subsequences) of π(t) = t _{π(1)} t _{π(2)} ⋯ t _{π(m)} (we call this ordering of the multiset, π(t), a permutation of t). We first show that in its most general setting the RCSstr problem is NP-complete and hard to approximate within a factor of n ^{1 - ε} , for any ε > 0, unless P = NP. Afterwards, we present two separate reductions to show that the RCSstr problem remains NP-Hard even in the case where the elements of t are drawn from a binary alphabet or for the case where all input strings are of length two. We then present some approximation results for several variants of the RCSstr problem. In the second part of this paper, we turn to the RCSseq problem, where we present some hardness results, tight lower bounds and approximation algorithms.

Original language | English |
---|---|

Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 467-478 |

Number of pages | 12 |

Volume | 6661 LNCS |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

Event | 22nd Annual Symposium on Combinatorial Pattern Matching, CPM 2011 - Palermo, Italy Duration: Jun 27 2011 → Jun 29 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 6661 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 22nd Annual Symposium on Combinatorial Pattern Matching, CPM 2011 |
---|---|

Country | Italy |

City | Palermo |

Period | 6/27/11 → 6/29/11 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 6661 LNCS, pp. 467-478). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6661 LNCS). https://doi.org/10.1007/978-3-642-21458-5_39

**Restricted common superstring and restricted common supersequence.** / Clifford, Raphaël; Gotthilf, Zvi; Lewenstein, Moshe; Popa, Alexandru.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 6661 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6661 LNCS, pp. 467-478, 22nd Annual Symposium on Combinatorial Pattern Matching, CPM 2011, Palermo, Italy, 6/27/11. https://doi.org/10.1007/978-3-642-21458-5_39

}

TY - GEN

T1 - Restricted common superstring and restricted common supersequence

AU - Clifford, Raphaël

AU - Gotthilf, Zvi

AU - Lewenstein, Moshe

AU - Popa, Alexandru

PY - 2011

Y1 - 2011

N2 - The shortest common superstring and the shortest common supersequence are two well studied problems having a wide range of applications. In this paper we consider both problems with resource constraints, denoted as the Restricted Common Superstring (shortly RCSstr) problem and the Restricted Common Supersequence (shortly RCSseq). In the RCSstr (RCSseq) problem we are given a set S of n strings, s 1, s 2, ..., s n , and a multiset t = {t 1, t 2, ..., t m }, and the goal is to find a permutation π: {1, ..., m} → {1, ..., m} to maximize the number of strings in S that are substrings (subsequences) of π(t) = t π(1) t π(2) ⋯ t π(m) (we call this ordering of the multiset, π(t), a permutation of t). We first show that in its most general setting the RCSstr problem is NP-complete and hard to approximate within a factor of n 1 - ε , for any ε > 0, unless P = NP. Afterwards, we present two separate reductions to show that the RCSstr problem remains NP-Hard even in the case where the elements of t are drawn from a binary alphabet or for the case where all input strings are of length two. We then present some approximation results for several variants of the RCSstr problem. In the second part of this paper, we turn to the RCSseq problem, where we present some hardness results, tight lower bounds and approximation algorithms.

AB - The shortest common superstring and the shortest common supersequence are two well studied problems having a wide range of applications. In this paper we consider both problems with resource constraints, denoted as the Restricted Common Superstring (shortly RCSstr) problem and the Restricted Common Supersequence (shortly RCSseq). In the RCSstr (RCSseq) problem we are given a set S of n strings, s 1, s 2, ..., s n , and a multiset t = {t 1, t 2, ..., t m }, and the goal is to find a permutation π: {1, ..., m} → {1, ..., m} to maximize the number of strings in S that are substrings (subsequences) of π(t) = t π(1) t π(2) ⋯ t π(m) (we call this ordering of the multiset, π(t), a permutation of t). We first show that in its most general setting the RCSstr problem is NP-complete and hard to approximate within a factor of n 1 - ε , for any ε > 0, unless P = NP. Afterwards, we present two separate reductions to show that the RCSstr problem remains NP-Hard even in the case where the elements of t are drawn from a binary alphabet or for the case where all input strings are of length two. We then present some approximation results for several variants of the RCSstr problem. In the second part of this paper, we turn to the RCSseq problem, where we present some hardness results, tight lower bounds and approximation algorithms.

UR - http://www.scopus.com/inward/record.url?scp=79960081879&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960081879&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-21458-5_39

DO - 10.1007/978-3-642-21458-5_39

M3 - Conference contribution

SN - 9783642214578

VL - 6661 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 467

EP - 478

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -