The Longest Run Subsequence Problem: Further Complexity Results

TL;DR

This paper explores the computational complexity of the Longest Run Subsequence problem, establishing new fixed-parameter tractability results, kernelization limitations, and hardness under specific constraints, advancing understanding of its algorithmic challenges.

Contribution

It demonstrates fixed-parameter tractability when parameterized by the number of runs and proves the non-existence of polynomial kernels for certain parameters, also showing APX-hardness under specific restrictions.

Findings

FPT when parameterized by the number of runs in a solution

No polynomial kernel when parameterized by alphabet size or number of runs

APX-hard for strings with each symbol appearing at most twice

Abstract

Longest Run Subsequence is a problem introduced recently in the context of the scaffolding phase of genome assembly (Schrinner et al., WABI 2020). The problem asks for a maximum length subsequence of a given string that contains at most one run for each symbol (a run is a maximum substring of consecutive identical symbols). The problem has been shown to be NP-hard and to be fixed-parameter tractable when the parameter is the size of the alphabet on which the input string is defined. In this paper we further investigate the complexity of the problem and we show that it is fixed-parameter tractable when it is parameterized by the number of runs in a solution, a smaller parameter. Moreover, we investigate the kernelization complexity of Longest Run Subsequence and we prove that it does not admit a polynomial kernel when parameterized by the size of the alphabet or by the number of runs. Finally, we consider the restriction of Longest Run Subsequence when each symbol has at most two occurrences in the input string and we show that it is APX-hard.