String Consensus Problems with Swaps and Substitutions

2025-07-25

AI Summary

This paper studies the string consensus problem under the adjacent-swap-and-substitution edit distance: given a set of equal-length strings, find a target string whose maximum (or sum of) distances to the input strings is at most a threshold (d) (or minimized). It introduces adjacent swaps into the parameterized framework of string consensus for the first time. For the max-distance version, we prove fixed-parameter tractability (FPT) with respect to (d) and design a parameterized algorithm combining dynamic programming with bounded search trees. For the sum-distance minimization version, we present the first polynomial-time exact algorithm. Our work extends classical consensus models—previously restricted to substitution, insertion, and deletion—to incorporate adjacent swaps, thereby resolving a previously NP-hard generalized consensus problem. Under the unit-cost swap-substitution metric, our algorithms achieve theoretical optimality in both time complexity and parameter dependence.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Multiagent Systems: Distributed Problem Solving

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 extsc{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.

Problem

• Generalizes Closest String problem with swaps and substitutions
• Proves FPT complexity for distance parameter d
• Solves sum-of-distances variant in polynomial time

Innovation

• FPT algorithm for swap and substitution distance
• Polynomial-time sum distance minimization algorithm
• Generalized closest string with adjacent swaps