Optimal LZ-End Parsing is Hard

TL;DR

This paper proves that finding the optimal LZ-End parsing with the fewest phrases is NP-complete, introduces a MAX-SAT formulation for it, and analyzes the approximation ratio of greedy parsing.

Contribution

It establishes the computational hardness of optimal LZ-End parsing and provides a MAX-SAT approach and approximation bounds, advancing understanding of this compression method.

Findings

Optimal LZ-End parsing decision problem is NP-complete.

A MAX-SAT formulation for the problem is provided.

The greedy parsing can be up to twice as large as the optimal.

Abstract

LZ-End is a variant of the well-known Lempel-Ziv parsing family such that each phrase of the parsing has a previous occurrence, with the additional constraint that the previous occurrence must end at the end of a previous phrase. LZ-End was initially proposed as a greedy parsing, where each phrase is determined greedily from left to right, as the longest factor that satisfies the above constraint~[Kreft & Navarro, 2010]. In this work, we consider an optimal LZ-End parsing that has the minimum number of phrases in such parsings. We show that a decision version of computing the optimal LZ-End parsing is NP-complete by showing a reduction from the vertex cover problem. Moreover, we give a MAX-SAT formulation for the optimal LZ-End parsing adapting an approach for computing various NP-hard repetitiveness measures recently presented by [Bannai et al., 2022]. We also consider the approximation ratio of the size of greedy LZ-End parsing to the size of the optimal LZ-End parsing, and give a lower bound of the ratio which asymptotically approaches $2$.