String Consensus Problems with Swaps and Substitutions

TL;DR

This paper explores a generalized string consensus problem allowing swaps and substitutions, proving fixed-parameter tractability for the maximum distance version and providing a polynomial-time solution for the sum-distance variant.

Contribution

It introduces a new variant of the string consensus problem with swaps, establishes its fixed-parameter tractability, and offers an efficient algorithm for the sum-distance version.

Findings

The generalized problem is NP-hard with only swaps.

The maximum distance version is fixed-parameter tractable.

A polynomial-time algorithm is provided for the sum-distance variant.

Abstract

String consensus problems aim at finding a string that minimizes some given distance with respect to an input set of strings. In particular, in the Closest string problem, we are given a set of strings of equal length and a radius $d$. The objective is to find a new string that differs from each input string by at most $d$ substitutions. We study a generalization of this problem where, in addition to substitutions, swaps of adjacent characters are also permitted, each operation incurring a unit cost. Amir et al. showed that this generalized problem is NP-hard, even when only swaps are allowed. In this paper, we show that it is FPT with respect to the parameter $d$. Moreover, we investigate a variant in which the goal is to minimize the sum of distances from the output string to all input strings. For this version, we present a polynomial-time algorithm.