## The Maximum Equality-Free String Factorization Problem: Gaps vs. No Gaps

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##### Date

2020

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Springer

##### Abstract

A factorization of a string w is a partition of w into substrings u1,…,uk such that w=u1u2⋯uk . 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 decide, for a given string w and integer k, whether w admits an equality-free factorization with k factors.
Equality-free factorizations have lately received attention because of their application in DNA self-assembly. Condon et al. (CPM 2012) study a version of the problem and show that it is NP -complete to decide if there exists an equality-free factorization with an upper bound on the length of the factors. At STACS 2015, Fernau et al. show that the maximum equality-free factorization problem with a lower bound on the number of factors is NP -complete. Shortly after, Schmid (CiE 2015) presents results concerning the Fixed Parameter Tractability of the problems.
In this paper we approach equality free factorizations from a practical point of view i.e. we wish to obtain good solutions on given instances. To this end, we provide approximation algorithms, heuristics, Integer Programming models, an improved FPT algorithm and we also conduct experiments to analyze the performance of our proposed algorithms.
Additionally, we study a relaxed version of the problem where gaps are allowed between factors and we design a constant factor approximation algorithm for this case. Surprisingly, after extensive experiments we conjecture that the relaxed problem has the same optimum as the original.

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##### Keywords

String factorization, Equality-free, String algorithms, Heuristics