Hardness and algorithmic results for the approximate cover problem

TL;DR

This paper advances the understanding of the approximate string cover problem by providing approximation algorithms, fixed-parameter tractability results, and complexity analyses for various metrics, including pseudometrics.

Contribution

It introduces a new approximation algorithm with ratio √OPT, an FPT algorithm for small alphabets, and explores the problem's complexity under pseudometrics.

Findings

Approximation algorithm with ratio √OPT.

FPT algorithm with respect to alphabet size.

Complexity analysis for pseudometrics and equivalence of Hamming distance over unbounded and fixed alphabets.

Abstract

In CPM 2017, Amir et al. introduce a problem, named \emph{approximate string cover} (\textbf{ACP}), motivated by many aplications including coding and automata theory, formal language theory, combinatorics and molecular biology. A \emph{cover} of a string $T$ is a string $C$ for which every letter of $T$ lies within some occurrence of $C$. The input of the \textbf{ACP} problem consists of a string $T$ and an integer $m$ (less than the length of $T$), and the goal is to find a string $C$ of length $m$ that covers a string $T'$ which is as close to $T$ as possible (under some predefined distance). Amir et al. study the problem for the Hamming distance. In this paper we continue the work of Amir et al. and show the following results: We show an approximation algorithm for the \textbf{ACP} with an approximation ratio of $\sqrt{OPT}$, where OPT is the size of the optimal solution. We provide an FPT algorithm with respect to the alphabet size. \item The \textbf{ACP} problem naturally extends to pseudometrics. Moreover, we show that for some family of pseudometrics, that we term \emph{homogenous additive pseudometrics}, the complexity of \textbf{ACP} remains unchanged. We partially give an answer to an open problem of Amir et al. and show that the Hamming distance over an unbounded alphabet is equivalent to an extended metric over a fixed sized alphabet.