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  1. A collection of sets displays aproximity gapwith respect to some property if for every set in the collection, either (i) all members areδ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members areδ-close to the property. In particular, no set in the collection has roughly half of its membersδ-close to the property and the othersδ-far from it.

    We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to anyδsmaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least)linearsize in the RS code dimension, forδsmaller than the unique decoding radius. Concretely, ifδis smaller than half the minimal distance of an RS code\(V\subset {\mathbb {F}}_q^n \), every affine space is either entirelyδ-close to the code, or alternatively at most an (n/q)-fraction of it isδ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems.

    We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on aformal elementof an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.

     
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    Free, publicly-accessible full text available August 15, 2024
  2. We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC 0 circuits for division, matrix powering, and related problems, where the improvement is in terms of “majority depth” (initially studied by Maciel and Thérien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy, and we answer a question posed by Yap.

     
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    Free, publicly-accessible full text available June 21, 2024
  3. Lane Hemaspaandra (Ed.)

    The end is near. More precisely, the end of my time as a (non-emeritus) professor is near. Thus I'm particularly grateful to Lane for inviting me to share some thoughts here, because there are a few things that I want to get off my chest. The Computational Complexity Column provides the perfect podium for this.

     
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  4. Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field F that outputs the evaluations of an m-variate polynomial of degree less than d in each variable at N points in time (dm + N)1+o(1) · poly(m, d, log |F|) for all m ∈ N and all sufficiently large d ∈ N. A previous work of Kedlaya and Umans (FOCS 2008, SICOMP 2011) achieved the same time complexity when the number of variables m is at most d^{o(1)} and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than d^{o(1)}. In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously knownalgorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski and Williams (IPEC 2017, Algorithmica 2019), and that of Bhargava, Ghosh, Kumar and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and in particular, avoids the use of the Bombieri–Vinogradov Theorem. However, it requires a mild assumption that the field size is bounded by an exponential-tower in d of bounded height. 
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  5. Ahn, Hee-Kap ; Sadakane, Kunihiko (Ed.)
    A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤^{NC^0}_m reductions. In this paper, we improve this, to show that the complement of MKTP is hard for the (apparently larger) class NISZK_L under not only ≤^{NC^0}_m reductions but even under projections. Also, the complement of MKTP is hard for NISZK under ≤^{P/poly}_m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK_L is the non-interactive version of the class SZK_L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP). 
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  6. Bojanczy, Mikolaj ; Chekuri, Chandra (Ed.)
    One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs. 
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  7. We survey recent developments related to the Minimum Circuit Size Problem and time-bounded Kolmogorov Complexity. 
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