Can You Solve Closest String Faster Than Exhaustive Search?

Amir Abboud, Nick Fischer, Elazar Goldenberg, C. S. Karthik, Ron Safier

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set X ⊆ Σd of n strings, find the string x minimizing the radius of the smallest Hamming ball around x that encloses all the strings in X. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: In the continuous Closest String problem, the goal is to find the solution string x anywhere in Σd. For binary strings, the exhaustive search algorithm runs in time O(2d poly(nd)) and we prove that it cannot be improved to time O(2(1−ϵ)d poly(nd)), for any ϵ > 0, unless the Strong Exponential Time Hypothesis fails. In the discrete Closest String problem, x is required to be in the input set X. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time n2±o(1) whenever the dimension is ω(log n) < d < no(1). We complement this known hardness result with new algorithms, proving essentially that whenever d falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-d regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in X.

Original languageEnglish
Title of host publication31st Annual European Symposium on Algorithms, ESA 2023
EditorsInge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772952
DOIs
StatePublished - Sep 2023
Event31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands
Duration: 4 Sep 20236 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume274
ISSN (Print)1868-8969

Conference

Conference31st Annual European Symposium on Algorithms, ESA 2023
Country/TerritoryNetherlands
CityAmsterdam
Period4/09/236/09/23

All Science Journal Classification (ASJC) codes

  • Software

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