124 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Can You Solve Closest String Faster Than Exhaustive Search?

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Quantum fine-grained complexity

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    Can You Solve Closest String Faster than Exhaustive Search?

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    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 XΣdX \subseteq \Sigma^d of nn strings, find the string xx^* minimizing the radius of the smallest Hamming ball around xx^* that encloses all the strings in XX. 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: \bullet In the continuous Closest String problem, the goal is to find the solution string xx^* anywhere in Σd\Sigma^d. For binary strings, the exhaustive search algorithm runs in time O(2dpoly(nd))O(2^d poly(nd)) and we prove that it cannot be improved to time O(2(1ϵ)dpoly(nd))O(2^{(1-\epsilon) d} poly(nd)), for any ϵ>0\epsilon > 0, unless the Strong Exponential Time Hypothesis fails. \bullet In the discrete Closest String problem, xx^* is required to be in the input set XX. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time n2±o(1)n^{2 \pm o(1)} whenever the dimension is ω(logn)<d<no(1)\omega(\log n) < d < n^{o(1)}. We complement this known hardness result with new algorithms, proving essentially that whenever dd falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-dd 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 XX

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    Improved Merlin-Arthur Protocols for Central Problems in Fine-Grained Complexity

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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