## Abstract

A factorization of a string S is a partition of w into substrings u1, . . ., u_{k} such that S = u1u2 · · · u_{k}. Such a partition is called equality-free if no two factors are equal: ui ≠ uj, ∀i, j with i ≠ j. The maximum equality-free factorization problem is to find for a given string S, the largest integer k for which S admits an equality-free factorization with k factors. Equality-free factorizations have lately received attention because of their applications in DNA self-assembly. The best approximation algorithm known for the problem is the natural greedy algorithm, that chooses iteratively from left to right the shortest factor that does not appear before. This algorithm has a √n approximation ratio (SOFSEM 2020) and it is an open problem whether there is a better solution. Our main result is to show that the natural greedy algorithm is a Θ(n^{1}/^{4}) approximation algorithm for the maximum equality-free factorization problem. Thus, we disprove one of the conjectures of Mincu and Popa (SOFSEM 2020) according to which the greedy algorithm is a Θ(√n) approximation. The most challenging part of the proof is to show that the greedy algorithm is an O(n^{1}/^{4}) approximation. We obtain this algorithm via prefix free factor families, i.e. a set of non-overlapping factors of the string which are pairwise non-prefixes of each other. In the paper we show the relation between prefix free factor families and the maximum equality-free factorization. Moreover, as a byproduct we present another approximation algorithm that achieves an approximation ratio of O(n^{1}/^{4}) that we believe is of independent interest and may lead to improved algorithms. We then show that the natural greedy algorithm has an approximation ratio that is Ω(n^{1}/^{4}) via a clever analysis which shows that the greedy algorithm is Θ(n^{1}/^{4}) for the maximum equality-free factorization problem.

Original language | English |
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Title of host publication | 34th Annual Symposium on Combinatorial Pattern Matching, CPM 2023 |

Editors | Laurent Bulteau, Zsuzsanna Liptak |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772761 |

DOIs | |

State | Published - Jun 2023 |

Event | 34th Annual Symposium on Combinatorial Pattern Matching, CPM 2023 - Marne-la-Vallee, France Duration: 26 Jun 2023 → 28 Jun 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 259 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 34th Annual Symposium on Combinatorial Pattern Matching, CPM 2023 |
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Country/Territory | France |

City | Marne-la-Vallee |

Period | 26/06/23 → 28/06/23 |

### Bibliographical note

Publisher Copyright:© 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

## Keywords

- NP-hard problem
- approximation algorithm
- string factorization