Journal of the ACM

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ISSN / EISSN : 0004-5411 / 1557-735X
Total articles ≅ 3,069
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Noah Fleming, Denis Pankratov, Toniann Pitassi, Robert Robere
Journal of the ACM; https://doi.org/10.1145/3486680

Abstract:
The random k -SAT model is one of the most important and well-studied distributions over k -SAT instances. It is closely connected to statistical physics and is a benchmark for satisfiability algorithms. We show that when k = Θ (log n ), any Cutting Planes refutation for random k -SAT requires exponential length in the regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.
Gadi Taubenfeld
Journal of the ACM; https://doi.org/10.1145/3529752

Abstract:
Assuming that there is an a priori agreement between processes on the names of shared memory locations, as is done in almost all the publications on concurrent shared memory algorithms, is tantamount to assuming that agreement has already been solved at a lower-level. It is intriguing to figure out how coordination can be achieved without relying on such lower-level agreement. To better understand the new model, we first design new algorithms for several important problems, such as mutual exclusion, consensus, election, and renaming. Then, we prove space lower bounds, impossibility results, and resolve two foundational long-standing open problems in the context of anonymous memory systems. Using these results, we identify fundamental differences between the standard shared memory model and the strictly weaker anonymous shared memory model. Besides enabling us to understand better the intrinsic limits for coordinating the actions of asynchronous processes, the new model has been shown to be useful in modeling biologically inspired distributed computing methods, especially those based on ideas from molecular biology.
Ran Raz, Avishay Tal
Journal of the ACM; https://doi.org/10.1145/3530258

Abstract:
We present a distribution \({\mathcal {D}} \) over inputs in { ± 1} 2 N , such that: (1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time O (log N ), that distinguishes between \({\mathcal {D}} \) and the uniform distribution with advantage \(\Omega(1/\log N)\) . (2) No Boolean circuit of quasi-polynomial size and constant depth distinguishes between \({\mathcal {D}} \) and the uniform distribution with advantage better than \({\mathrm{polylog}}(N)/\sqrt {N} \) . By well-known reductions, this gives a separation of the classes Promise- BQP and Promise- PH in the black-box model and implies an oracle relative to which BQP is not contained in PH .
David Harvey, Joris van der Hoeven
Journal of the ACM, Volume 69, pp 1-40; https://doi.org/10.1145/3505584

Abstract:
Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than \( n \) over a finite field \( \mathbb {F}_q \) with \( q \) elements can be multiplied in time \( O (n \log q \log (n \log q)) \) , uniformly in \( q \) . Under the same hypothesis, we show how to multiply two \( n \) -bit integers in time \( O (n \log n) \) ; this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [ 22 ]. Our results hold in the Turing machine model with a finite number of tapes.
Journal of the ACM, Volume 69, pp 1-44; https://doi.org/10.1145/3511100

Abstract:
Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP , as long as one is willing to make a suitable physical assumption : if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP * model captures this setting. In this work, we study the following question: Does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP * . Our proof consists of constructing a zero knowledge interactive probabilistically checkable proof with a strong algebraic structure, and then lifting it to the MIP * model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.
Journal of the ACM, Volume 69, pp 1-34; https://doi.org/10.1145/3502264

Abstract:
We study the atomic embeddability testing problem, which is a common generalization of clustered planarity ( c-planarity , for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time. Our algorithm for atomic embeddability combines ideas from Carmesin’s work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory. Finally, we give a polynomial-time reduction from atomic embeddability to thickenability thereby showing that both problems are polynomially equivalent, and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.
, Rajesh Jayaram, David P. Woodruff, Eylon Yogev
Journal of the ACM, Volume 69, pp 1-33; https://doi.org/10.1145/3498334

Abstract:
We investigate the adversarial robustness of streaming algorithms. In this context, an algorithm is considered robust if its performance guarantees hold even if the stream is chosen adaptively by an adversary that observes the outputs of the algorithm along the stream and can react in an online manner. While deterministic streaming algorithms are inherently robust, many central problems in the streaming literature do not admit sublinear-space deterministic algorithms; on the other hand, classical space-efficient randomized algorithms for these problems are generally not adversarially robust. This raises the natural question of whether there exist efficient adversarially robust (randomized) streaming algorithms for these problems. In this work, we show that the answer is positive for various important streaming problems in the insertion-only model, including distinct elements and more generally F p -estimation, F p -heavy hitters, entropy estimation, and others. For all of these problems, we develop adversarially robust (1+ε)-approximation algorithms whose required space matches that of the best known non-robust algorithms up to a poly(log n , 1/ε) multiplicative factor (and in some cases even up to a constant factor). Towards this end, we develop several generic tools allowing one to efficiently transform a non-robust streaming algorithm into a robust one in various scenarios.
Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, Fabio Zanasi
Journal of the ACM, Volume 69, pp 1-58; https://doi.org/10.1145/3502719

Abstract:
String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.
Venkatesan Guruswami, Chaoping Xing
Journal of the ACM, Volume 69, pp 1-48; https://doi.org/10.1145/3506668

Abstract:
We give new constructions of two classes of algebraic code families that are efficiently list decodable with small output list size from a fraction 1-R-ε of adversarial errors, where R is the rate of the code, for any desired positive constant ε. The alphabet size depends only ε and is nearly optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield . In both cases, we develop a linear-algebraic approach to perform list decoding, which pins down the candidate messages to a subspace with a nice “periodic” structure. To prune this subspace and obtain a good bound on the list size, we pick subcodes of these codes by pre-coding into certain subspace-evasive sets that are guaranteed to have small intersection with the sort of periodic subspaces that arise in our list decoding. We develop two approaches for constructing such subspace-evasive sets. The first is a Monte Carlo construction of hierearchical subspace-evasive (h.s.e.) sets that leads to excellent list size but is not explicit. The second approach exploits a further ultra-periodicity of our subspaces and uses a novel construct called subspace designs , which were subsequently constructed explicitly and also found further applications in pseudorandomness. To get a family of codes over a fixed alphabet size, we instantiate our approach with algebraic-geometric codes based on the Garcia–Stichtenoth tower of function fields. Combining this with pruning via h.s.e. sets yields codes list-decodable up to a 1-R-ε error fraction with list size bounded by O (1/ε), matching the existential bound for random codes up to constant factors. Further, the alphabet size can be made exp ( Õ (1/ε 2 )), which is not much worse than the lower bound of exp (Ω (1/ε)). The parameters we achieve are thus quite close to the existential bounds in all three aspects (error-correction radius, alphabet size, and list size) simultaneously. This construction is, however, Monte Carlo and the claimed list-decoding property only holds with high probability. Once the code is (efficiently) sampled, the encoding/decoding algorithms are deterministic with a running time O _ε ( N c ) for an absolute constant c , where N is the code’s block length. Using subspace designs instead for the pruning, our approach yields the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-ε fraction of errors over an alphabet of constant size exp (Õ(1/ε 2 )). The list-size bound is upper bounded by a very slowly growing function of the block length N ; in particular, it is at most O(log ( r ) N ) (the r th iterated logarithm) for any fixed integer r . The explicit construction avoids the shortcoming of the Monte Carlo sampling at the expense of a slightly worse list size.
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