String factorisations with maximum or minimum dimension

TL;DR

This paper investigates string factorisation problems focusing on bounding factor length and maximizing or minimizing the number of distinct factors, providing complexity results and approximation algorithms for these NP-hard problems.

Contribution

It introduces the FMD problem, analyzes its complexity, and offers approximation algorithms and exact solutions for string factorisation with bounded factor length.

Findings

FmD is NP-hard for k=2 with a 3/2-approximation algorithm.

FMD is NP-hard for all k ≥ 3.

Proposed a 2-approximation algorithm and an exact exponential algorithm.

Abstract

In this paper we consider two problems concerning string factorisation. Specifically given a string $w$ and an integer $k$ find a factorisation of $w$ where each factor has length bounded by $k$ and has the minimum (the FmD problem) or the maximum (the FMD problem) number of different factors. The FmD has been proved to be NP-hard even if $k=2$ in [9] and for this case we provide a $3/2$-approximation algorithm. The FMD problem, up to our knowledge has not been considered in the literature. We show that this problem is NP-hard for any $k\geq 3$. In view of this we propose a $2$-approximation algorithm (for any $k$) an exact exponential algorithm. We conclude with some open problems.