TY - GEN

T1 - Can You Solve Closest String Faster Than Exhaustive Search?

AU - Abboud, Amir

AU - Fischer, Nick

AU - Goldenberg, Elazar

AU - Karthik, C. S.

AU - Safier, Ron

N1 - Publisher Copyright: © Amir Abboud, Nick Fischer, Elazar Goldenberg, Karthik C. S., and Ron Safier.

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85173435319&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ESA.2023.3

DO - https://doi.org/10.4230/LIPIcs.ESA.2023.3

M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 31st Annual European Symposium on Algorithms, ESA 2023

A2 - Li Gortz, Inge

A2 - Farach-Colton, Martin

A2 - Puglisi, Simon J.

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 31st Annual European Symposium on Algorithms, ESA 2023

Y2 - 4 September 2023 through 6 September 2023

ER -