Subjects -> MATHEMATICS (Total: 1013 journals)
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MATHEMATICS (714 journals)            First | 1 2 3 4     

Showing 601 - 538 of 538 Journals sorted alphabetically
Research in the Mathematical Sciences     Open Access  
Research Journal of Pure Algebra     Open Access   (Followers: 1)
Researches in Mathematics     Open Access  
Results in Control and Optimization     Open Access  
Results in Mathematics     Hybrid Journal  
Results in Nonlinear Analysis     Open Access  
Review of Symbolic Logic     Full-text available via subscription   (Followers: 2)
Reviews in Mathematical Physics     Hybrid Journal   (Followers: 1)
Revista Baiana de Educação Matemática     Open Access  
Revista Bases de la Ciencia     Open Access  
Revista BoEM - Boletim online de Educação Matemática     Open Access  
Revista Colombiana de Matemáticas     Open Access   (Followers: 1)
Revista de Ciencias     Open Access  
Revista de Educación Matemática     Open Access  
Revista de la Escuela de Perfeccionamiento en Investigación Operativa     Open Access  
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas     Partially Free  
Revista de Matemática : Teoría y Aplicaciones     Open Access   (Followers: 1)
Revista Digital: Matemática, Educación e Internet     Open Access  
Revista Electrónica de Conocimientos, Saberes y Prácticas     Open Access  
Revista Integración : Temas de Matemáticas     Open Access  
Revista Internacional de Sistemas     Open Access  
Revista Latinoamericana de Etnomatemática     Open Access  
Revista Latinoamericana de Investigación en Matemática Educativa     Open Access  
Revista Matemática Complutense     Hybrid Journal  
Revista REAMEC : Rede Amazônica de Educação em Ciências e Matemática     Open Access  
Revista SIGMA     Open Access  
Ricerche di Matematica     Hybrid Journal  
RMS : Research in Mathematics & Statistics     Open Access  
Royal Society Open Science     Open Access   (Followers: 7)
Russian Journal of Mathematical Physics     Full-text available via subscription  
Russian Mathematics     Hybrid Journal  
Sahand Communications in Mathematical Analysis     Open Access  
Sampling Theory, Signal Processing, and Data Analysis     Hybrid Journal  
São Paulo Journal of Mathematical Sciences     Hybrid Journal  
Science China Mathematics     Hybrid Journal   (Followers: 1)
Science Progress     Full-text available via subscription   (Followers: 1)
Sciences & Technologie A : sciences exactes     Open Access  
Selecta Mathematica     Hybrid Journal   (Followers: 1)
SeMA Journal     Hybrid Journal  
Semigroup Forum     Hybrid Journal   (Followers: 1)
Set-Valued and Variational Analysis     Hybrid Journal  
SIAM Journal on Applied Mathematics     Hybrid Journal   (Followers: 11)
SIAM Journal on Computing     Hybrid Journal   (Followers: 11)
SIAM Journal on Control and Optimization     Hybrid Journal   (Followers: 18)
SIAM Journal on Discrete Mathematics     Hybrid Journal   (Followers: 8)
SIAM Journal on Financial Mathematics     Hybrid Journal   (Followers: 3)
SIAM Journal on Mathematics of Data Science     Hybrid Journal   (Followers: 1)
SIAM Journal on Matrix Analysis and Applications     Hybrid Journal   (Followers: 3)
SIAM Journal on Optimization     Hybrid Journal   (Followers: 12)
Siberian Advances in Mathematics     Hybrid Journal  
Siberian Mathematical Journal     Hybrid Journal  
Sigmae     Open Access  
SILICON     Hybrid Journal  
SN Partial Differential Equations and Applications     Hybrid Journal  
Soft Computing     Hybrid Journal   (Followers: 7)
Statistics and Computing     Hybrid Journal   (Followers: 14)
Stochastic Analysis and Applications     Hybrid Journal   (Followers: 3)
Stochastic Partial Differential Equations : Analysis and Computations     Hybrid Journal   (Followers: 2)
Stochastic Processes and their Applications     Hybrid Journal   (Followers: 6)
Stochastics and Dynamics     Hybrid Journal   (Followers: 2)
Studia Scientiarum Mathematicarum Hungarica     Full-text available via subscription   (Followers: 1)
Studia Universitatis Babeș-Bolyai Informatica     Open Access  
Studies In Applied Mathematics     Hybrid Journal   (Followers: 1)
Studies in Mathematical Sciences     Open Access   (Followers: 1)
Superficies y vacio     Open Access  
Suska Journal of Mathematics Education     Open Access   (Followers: 1)
Swiss Journal of Geosciences     Hybrid Journal   (Followers: 1)
Synthesis Lectures on Algorithms and Software in Engineering     Full-text available via subscription   (Followers: 2)
Synthesis Lectures on Mathematics and Statistics     Full-text available via subscription   (Followers: 1)
Tamkang Journal of Mathematics     Open Access  
Tatra Mountains Mathematical Publications     Open Access  
Teaching Mathematics     Full-text available via subscription   (Followers: 10)
Teaching Mathematics and its Applications: An International Journal of the IMA     Hybrid Journal   (Followers: 4)
Teaching Statistics     Hybrid Journal   (Followers: 8)
Technometrics     Full-text available via subscription   (Followers: 8)
The Journal of Supercomputing     Hybrid Journal   (Followers: 1)
The Mathematica journal     Open Access  
The Mathematical Gazette     Full-text available via subscription   (Followers: 1)
The Mathematical Intelligencer     Hybrid Journal  
The Ramanujan Journal     Hybrid Journal  
The VLDB Journal     Hybrid Journal   (Followers: 2)
Theoretical and Mathematical Physics     Hybrid Journal   (Followers: 7)
Theory and Applications of Graphs     Open Access  
Topological Methods in Nonlinear Analysis     Full-text available via subscription  
Transactions of the London Mathematical Society     Open Access   (Followers: 1)
Transformation Groups     Hybrid Journal  
Turkish Journal of Mathematics     Open Access  
Ukrainian Mathematical Journal     Hybrid Journal  
Uniciencia     Open Access  
Uniform Distribution Theory     Open Access  
Unisda Journal of Mathematics and Computer Science     Open Access  
Unnes Journal of Mathematics     Open Access   (Followers: 1)
Unnes Journal of Mathematics Education     Open Access   (Followers: 2)
Unnes Journal of Mathematics Education Research     Open Access   (Followers: 1)
Ural Mathematical Journal     Open Access  
Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki     Open Access  
Vestnik St. Petersburg University: Mathematics     Hybrid Journal  
VFAST Transactions on Mathematics     Open Access   (Followers: 1)
Vietnam Journal of Mathematics     Hybrid Journal  
Vinculum     Full-text available via subscription  
Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics     Open Access   (Followers: 2)
Water SA     Open Access   (Followers: 1)
Water Waves     Hybrid Journal  
Zamm-Zeitschrift Fuer Angewandte Mathematik Und Mechanik     Hybrid Journal   (Followers: 1)
ZDM     Hybrid Journal   (Followers: 2)
Zeitschrift für angewandte Mathematik und Physik     Hybrid Journal   (Followers: 2)
Zeitschrift fur Energiewirtschaft     Hybrid Journal  
Zetetike     Open Access  

  First | 1 2 3 4     

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SIAM Journal on Computing
Journal Prestige (SJR): 0.874
Citation Impact (citeScore): 2
Number of Followers: 11  
 
  Hybrid Journal Hybrid journal (It can contain Open Access articles)
ISSN (Print) 0097-5397 - ISSN (Online) 1095-7111
Published by Society for Industrial and Applied Mathematics Homepage  [17 journals]
  • Approximately Counting and Sampling Small Witnesses Using a Colorful
           Decision Oracle

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      Authors: Holger Dell, John Lapinskas, Kitty Meeks
      Pages: 849 - 899
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 849-899, August 2022.
      In this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly and can be accessed only through its colorful independence oracle: The colorful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colorful with respect to a given vertex-coloring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. [ACM Trans. Algorithms, 16 (2020), 52], Dell and Lapinskas [Proceedings of STOC, ACM, 2018, pp. 281--288], and Bhattacharya et al. [Proceedings of ISAAC, 2019]. Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as $k$-SUM, $k$-OV, and weighted $k$-Clique) and parameterized complexity (such as induced subgraphs of size $k$ or weight-$k$ solutions to constraint satisfaction problems).
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-05T07:00:00Z
      DOI: 10.1137/19M130604X
      Issue No: Vol. 51, No. 4 (2022)
       
  • A Local Search Framework for Experimental Design

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      Authors: Lap Chi Lau, Hong Zhou
      Pages: 900 - 951
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 900-951, August 2022.
      We present a local search framework to design and analyze both combinatorial algorithms and rounding algorithms for experimental design problems. This framework provides a unifying approach to match and improve all known results in D/A/E-design and to obtain new results in previously unknown settings. For combinatorial algorithms, we provide a new analysis of the classical Fedorov's exchange method. We prove that this simple local search algorithm works well as long as there exists an almost optimal solution with good condition number. Moreover, we design a new combinatorial local search algorithm for E-design using the regret minimization framework. For rounding algorithms, we provide a unified randomized exchange algorithm to match and improve previous results for D/A/E-design. Furthermore, the algorithm works in the more general setting to approximately satisfy multiple knapsack constraints, which can be used for weighted experimental design and for incorporating fairness constraints into experimental design.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-07T07:00:00Z
      DOI: 10.1137/20M1386542
      Issue No: Vol. 51, No. 4 (2022)
       
  • Differentially Private Learning of Geometric Concepts

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      Authors: Haim Kaplan, Yishay Mansour, Yossi Matias, Uri Stemmer
      Pages: 952 - 974
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 952-974, August 2022.
      We present efficient differentially private algorithms for learning unions of polygons in the plane (which are not necessarily convex). Our algorithms are $(\alpha,\beta)$--probably approximately correct and $(\varepsilon,\delta)$--differentially private using a sample of size $\tilde{O}\left(\frac{1}{\alpha\varepsilon}k\log d\right)$, where the domain is $[d]\times[d]$ and $k$ is the number of edges in the union of polygons. Our algorithms are obtained by designing a private variant of the classical (nonprivate) learner for conjunctions using the greedy algorithm for set cover.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-07T07:00:00Z
      DOI: 10.1137/21M1406428
      Issue No: Vol. 51, No. 4 (2022)
       
  • Caching with Time Windows and Delays

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      Authors: Anupam Gupta, Amit Kumar, Debmalya Panigrahi
      Pages: 975 - 1017
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 975-1017, August 2022.
      We consider two generalizations of the classical weighted paging problem that incorporate the notion of delayed service of page requests. The first is the (weighted) paging with time windows (\sf PageTW) problem, which is like the classical weighted paging problem except that each page request only needs to be served before a given deadline. This problem arises in many practical applications of online caching, such as the “deadline” I/O scheduler in the Linux kernel and video-on-demand streaming. The second, and more general, problem is the (weighted) paging with delay (\sf PageD) problem, where the delay in serving a page request results in a penalty being added to the objective. This problem generalizes the caching problem to allow delayed service, a line of work that has recently gained traction in online algorithms (e.g., [Y. Emek, S. Kutten, and R. Wattenhofer, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, 2016, pp. 333--344; Y. Azar et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 551--563; Y. Azar and N. Touitou, Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science, 2019, pp. 60--71]). We give $O(\log k\log n)$-competitive algorithms for both the \sf PageTW and \sf PageD problems on $n$ pages with a cache of size $k$. This significantly improves on the previous best bounds of $O(k)$ for both problems [Y. Azar et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 551--563]. We also consider the offline \sf PageTW and \sf PageD problems, for which we give $O(1)$-approximation algorithms and prove APX-hardness. These are the first results for the offline problems; even NP-hardness was not known before our work. At the heart of our algorithms is a novel “hitting-set” LP relaxation of the \sf PageTW problem that overcomes the $\Omega(k)$ integrality gap of the natural LP for the problem. To the best of our knowledge, this is the first example of an LP-based algorithm for an online problem with delays/deadlines.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-07T07:00:00Z
      DOI: 10.1137/20M1346286
      Issue No: Vol. 51, No. 4 (2022)
       
  • A Spectral Approach to Network Design

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      Authors: Lap Chi Lau, Hong Zhou
      Pages: 1018 - 1064
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1018-1064, August 2022.
      We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional nonnegative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for the classical survivable network design problem, and partially answers a question of Bansal about survivable network design with concentration property. We also show many other applications of the spectral rounding results, including weighted experimental design and spectral network design.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-13T07:00:00Z
      DOI: 10.1137/20M1330762
      Issue No: Vol. 51, No. 4 (2022)
       
  • On Ray Shooting for Triangles in 3-Space and Related Problems

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      Authors: Esther Ezra, Micha Sharir
      Pages: 1065 - 1095
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1065-1095, August 2022.
      We consider several intersection searching problems that involve lines in ${\mathbb R}^3$ and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in ${\mathbb R}^3$, (ii) reporting intersections between query lines (segments, or rays) and input triangles in ${\mathbb R}^3$, as well as approximately counting the number of such intersections, (iii) computing the intersection of two nonconvex polyhedra in ${\mathbb R}^3$, (iv) detecting, counting, or reporting intersections in a set of lines in ${\mathbb R}^3$, and (v) output-sensitive construction of an arrangement of triangles in ${\mathbb R}^3$. Our approach is based on the polynomial partitioning technique. Our ray-shooting algorithm processes a set of $n$ triangles in ${\mathbb R}^3$ into a data structure for answering ray-shooting queries amid the given triangles, which uses $O(n^{3/2+{\varepsilon}})$ storage and expected preprocessing time, and answers a query in $O(n^{1/2+{\varepsilon}})$ time, for any ${\varepsilon}>0$. This is a significant improvement over known results, obtained more than 25 years ago, in which, with this amount of storage, the query time bound is roughly $n^{5/8}$. The algorithms for the other problems have similar performance bounds, with similar improvements over previous results. We also derive a nontrivial improved tradeoff between storage and query time. Using it, we obtain algorithms that answer $m$ queries on $n$ objects in $\max \left\{ O(m^{2/3}n^{5/6+{\varepsilon}} + n^{1+{\varepsilon}}),\; O(m^{5/6+{\varepsilon}}n^{2/3} + m^{1+{\varepsilon}}) \right\}$ expected time for any ${\varepsilon}>0$.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-14T07:00:00Z
      DOI: 10.1137/21M1408245
      Issue No: Vol. 51, No. 4 (2022)
       
  • Contextual Search via Intrinsic Volumes

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      Authors: Renato Paes Leme, Jon Schneider
      Pages: 1096 - 1125
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1096-1125, August 2022.
      We study the problem of contextual search, a multidimensional generalization of binary search that captures many problems in contextual decision-making. In contextual search, a learner is trying to learn the value of a hidden vector $v \in [0,1]^d$. Every round the learner is provided an adversarially chosen context $u_t \in \R^d$, submits a guess $p_t$ for the value of $\langle u_t, v\rangle$, learns whether $p_t < \langle u_t, v\rangle$, and incurs loss $\ell(\langle u_t, v\rangle, p_t)$ (for some loss function $\ell$). The learner's goal is to minimize their total loss over the course of $T$ rounds. We present an algorithm for the contextual search problem for the symmetric loss function $\ell(\theta, p) = \theta - p $ that achieves $O_{d}(1)$ total loss. We present a new algorithm for the dynamic pricing problem (which can be realized as a special case of the contextual search problem) that achieves $O_{d}(\log \log T)$ total loss, improving on the previous best known upper bounds of $O_{d}(\log T)$ and matching the known lower bounds (up to a polynomial dependence on $d$). Both algorithms make significant use of ideas from the field of integral geometry, most notably the notion of intrinsic volumes of a convex set. To the best of our knowledge, this is the first application of intrinsic volumes to algorithm design.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-18T07:00:00Z
      DOI: 10.1137/20M1385718
      Issue No: Vol. 51, No. 4 (2022)
       
  • Tighter Bounds on MultiParty Coin Flipping via Augmented Weak Martingales
           and Differentially Private Sampling

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      Authors: Amos Beimel, Iftach Haitner, Nikolaos Makriyannis, Eran Omri
      Pages: 1126 - 1171
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1126-1171, August 2022.
      In his seminal work, Cleve [Proceedings of the 18th Annual ACM Symposium on Theory of Computing, 1986, pp. 364--369] has proved that any $r$-round coin-flipping protocol can be efficiently biased by $\Theta(1/r)$. This lower bound was met for the two-party case by Moran, Naor, and Segev [J. Cryptology, 29 (2016), pp. 491--513] and the three-party case (up to a ${polylog}$ factor) by Haitner and Tsfadia [SIAM J. Comput., 46 (2017), pp. 479--542] and was approached for $n$-party protocols when $n< {loglog} r$ by Buchbinder et al. [Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, 2017, pp. 2580--2600]. For $n> {loglog} r$, however, the best bias for $n$-party coin-flipping protocols remains $O(n/\sqrt{r})$ achieved by the majority protocol of Awerbuch et al. [How to implement Bracha's ${O}(\log n)$ Byzantine Agreement Algorithm, manuscript, 1985]. Our main result is a tighter lower bound on the bias of coin-flipping protocols, showing that, for every constant $\varepsilon>0$, an $r^{\varepsilon}$-party $r$-round coin-flipping protocol can be efficiently biased by $\widetilde{\Omega}(1/\sqrt{r})$. As far as we know, this is the first improvement of Cleve's bound and is only $n=r^{\varepsilon}$ (multiplicative) far from the aforementioned upper bound of Awerbuch et al. We prove the above bound using two new results that we believe are of independent interest. The first result is that a sequence of (``augmented'') weak martingales have large gap: with constant probability there exists two adjacent variables whose gap is at least the ratio between the gap between the first and last variables and the square root of the number of variables. This generalizes over the result of Cleve and Impagliazzo [Martingales, Collective Coin Flipping and Discrete Control Processes (Extended Abstract), http://citeseerx.ist.psu.edu/viewdoc/summary'doi=10.1.1.51.1797, 1993], who showed that the above holds for strong martingales, and allows in some setting to exploit this gap by efficient algorithms. We prove the above using a novel argument that does not follow the more complicated approach of R. Cleve and R. Impagliazzo. The second result is a new sampling algorithm that uses a differentially private mechanism to minimize the effect of data divergence.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-21T07:00:00Z
      DOI: 10.1137/18M1210782
      Issue No: Vol. 51, No. 4 (2022)
       
  • Classical Verification of Quantum Computations

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      Authors: Urmila Mahadev
      Pages: 1172 - 1229
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1172-1229, August 2022.
      We present the first protocol allowing a classical computer to interactively verify the result of an efficient quantum computation. We achieve this by constructing a measurement protocol, which enables a classical verifier to use a quantum prover as a trusted measurement device. The protocol forces the prover to behave as follows: the prover must construct an $n$ qubit state of his choice, measure each qubit in the Hadamard or standard basis as directed by the verifier, and report the measurement results to the verifier. The soundness of this protocol is enforced based on the assumption that the learning with errors problem is computationally intractable for efficient quantum machines.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-21T07:00:00Z
      DOI: 10.1137/20M1371828
      Issue No: Vol. 51, No. 4 (2022)
       
  • Testing Linear-Invariant Properties

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      Authors: Jonathan Tidor, Yufei Zhao
      Pages: 1230 - 1279
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1230-1279, August 2022.
      We study the property testing of functions $\mathbb F_p^n\to[R]$ for fixed prime $p$ and positive integer $R$. We work in the natural model where we are allowed to query the function on a random subspace of constant dimension. We say that a property is testable if queries of this form can detect the property with one-sided error. Furthermore, a property is proximity oblivious-testable (PO-testable) if the test is also independent of the proximity parameter $\epsilon$. It is known that a number of natural properties such as linearity and being a low degree polynomial are PO-testable. These properties are examples of linear-invariant properties, meaning that they are preserved under linear automorphisms of the domain. Following work of Kaufman and Sudan, the study of linear-invariant properties has been an important problem in arithmetic property testing. A central conjecture in this field, proposed by Bhattacharyya, Grigorescu, and Shapira, is that a linear-invariant property is testable if and only if it is semi-subspace-hereditary. We prove two results; the first resolves this conjecture and the second classifies PO-testable properties: (1) A linear-invariant property is testable if and only if it is semi-subspace-hereditary. (2) A linear-invariant property is PO-testable if and only if it is locally characterized. Our innovations are twofold. We give a more powerful version of the compactness argument first introduced by Alon and Shapira. This relies on a new strong arithmetic regularity lemma in which one mixes different levels of Gowers uniformity. This allows us to extend the work of Bhattacharyya, Fischer, Hatami, Hatami, and Lovett by removing the bounded complexity restriction in their work. Our second innovation is a novel recoloring technique called patching that builds on earlier work by the authors and Fox. This Ramsey-theoretic technique is critical for working in the linear-invariant setting and allows us to remove the translation-invariant restriction present in previous work.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-21T07:00:00Z
      DOI: 10.1137/21M1397246
      Issue No: Vol. 51, No. 4 (2022)
       
  • Perfect Sampling in Infinite Spin Systems Via Strong Spatial Mixing

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      Authors: Konrad Anand, Mark Jerrum
      Pages: 1280 - 1295
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1280-1295, August 2022.
      We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph $G$. The sampling algorithm assumes strong spatial mixing together with subexponential growth of $G$. It produces a finite window onto a perfect sample from the Gibbs distribution. The run-time is linear in the size of the window, in particular it is constant for each vertex.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-26T07:00:00Z
      DOI: 10.1137/21M1437433
      Issue No: Vol. 51, No. 4 (2022)
       
  • A Near-Optimal Algorithm for Shortest Paths Among Curved Obstacles in the
           Plane

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      Authors: John Hershberger, Subhash Suri, Hakan Yildiz
      Pages: 1296 - 1340
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1296-1340, January 2022.
      We propose an algorithm for the problem of computing shortest paths among curved obstacles in the plane. If the obstacles have $O(n)$ description complexity, then the algorithm runs in $O(n\log n)$ time plus a term dependent on the properties of the boundary arcs. Specifically, if the arcs allow a certain kind of bisector intersection to be computed in constant time, or even in $O(\log n)$ time, then the running time of the overall algorithm is $O(n \log n)$. If the arcs support only constant-time tangent, intersection, and length queries, as is customarily assumed, then the algorithm computes an approximate shortest path, with relative error $\varepsilon$, in time $O(n\log n + n\log \frac{1}{\varepsilon})$. In fact, the algorithm computes an approximate shortest path map, a data structure with $O(n\log n)$ size, that allows it to report the (approximate) length of a shortest path from a fixed source point to any query point in the plane in $O(\log n)$ time. By applying an idea due to Wang [Proceedings of the $32$nd Annual ACM-SIAM Symposium on Discrete Algorithms, 2021, pp. 810--821], the algorithm's working storage and the size of the approximate shortest path map can be reduced to $O(n)$.
      Citation: SIAM Journal on Computing
      PubDate: 2022-08-09T07:00:00Z
      DOI: 10.1137/21M1428248
      Issue No: Vol. 51, No. 4 (2022)
       
  • Approximating Longest Common Subsequence in Linear Time: Beating the
           $\sqrt{{n}}$ Barrier

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      Authors: MohammadTaghi HajiAghayi, Masoud Seddighin, Saeedreza Seddighin, Xiaorui Sun
      Pages: 1341 - 1367
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1341-1367, August 2022.
      Longest common subsequence (LCS) is one of the most fundamental problems in combinatorial optimization. Apart from theoretical importance, LCS has enormous applications in bioinformatics, revision control systems, and data comparison programs. Although a simple dynamic program computes LCS in quadratic time, it has been recently proven that the problem admits a conditional lower bound and may not be solved in truly subquadratic time [2, 18]. In addition to this, LCS is notoriously hard with respect to approximation algorithms. Apart from a trivial sampling technique that obtains an $n^{x}$ approximation solution in time $O(n^{2-2x})$ nothing else is known for LCS. This is in sharp contrast to its dual problem edit distance, for which several linear time solutions were obtained in the past two decades [5, 6, 10, 11, 19]. In this work, we present the first nontrivial algorithm for approximating LCS in linear time. Our main result is a linear time algorithm for the LCS which has an approximation factor of $O(n^{0.4991319})$ in expectation. This beats the $\sqrt{n}$ barrier for approximating LCS in linear time.
      Citation: SIAM Journal on Computing
      PubDate: 2022-08-18T07:00:00Z
      DOI: 10.1137/19M1272068
      Issue No: Vol. 51, No. 4 (2022)
       
  • On CDCL-Based Proof Systems with the Ordered Decision Strategy

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      Authors: Nathan Mull, Shuo Pang, Alexander Razborov
      Pages: 1368 - 1399
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1368-1399, August 2022.
      We prove that conflict-driven clause learning (CDCL) SAT-solvers with the ordered decision strategy and the DECISION learning scheme are equivalent to ordered resolution. We also prove that by replacing this learning scheme with its opposite, which stops after the first nonconflict clause when backtracking, they become equivalent to general resolution. To the best of our knowledge, along with [M. Vinyals, “Hard Examples for Common Variable Decision Heuristics,” in Proceedings of the 34th AAAI Conference on Artificial Intelligence, 2020, pp. 1652--1659], this is the first theoretical study of the interplay between specific decision strategies and clause learning. For both results, we allow nondeterminism in the solver's ability to perform unit propagation, conflict analysis, and restarts, in a way that is similar to previous works in the literature. To aid the presentation of our results, and possibly future research, we define a model and language for discussing CDCL-based proof systems that allow for succinct and precise theorem statements.
      Citation: SIAM Journal on Computing
      PubDate: 2022-08-23T07:00:00Z
      DOI: 10.1137/20M1362528
      Issue No: Vol. 51, No. 4 (2022)
       
  • QMA-Hardness of Consistency of Local Density Matrices with Applications to
           Quantum Zero-Knowledge

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      Authors: Anne Broadbent, Alex Bredariol Grilo
      Pages: 1400 - 1450
      Abstract: SIAM Journal on Computing, Volume 51, Issue 4, Page 1400-1450, August 2022.
      We provide several advances to the understanding of the class of quantum Merlin--Arthur (QMA) proof systems, the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the consistency of local density matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most $k$ qubits, and the problem asks if there is an $n$-qubit global quantum state that is locally consistent with all of the $k$-qubit local density matrices. The containment of this problem in QMA and the QMA-hardness under Turing reductions were proved by Liu [in Proceeding of APPROX-RANDOM, 2006, pp. 438--449]. Liu also conjectured that CLDM is QMA-hard under Karp reductions, which is desirable for applications, and we finally prove this conjecture. We establish this result using the techniques of simulatable codes of Grilo, Slofstra, and Yuen [in Proceeding of FOCS, 2019, pp. 611--635], simplifying their proofs and tailoring them to the context of QMA. In order to develop applications of CLDM, we propose a framework that we call locally simulatable proofs for QMA: this provides QMA proofs that can be efficiently verified by probing only $k$ qubits and, furthermore, the reduced density matrix of any $k$-qubit subsystem of an accepting witness can be computed in polynomial time, independently of the witness. Within this framework, we show several advances in zero-knowledge in the quantum setting. We show for the first time a commit-and-open computational zero-knowledge proof system for all of QMA, as a quantum analogue of a “sigma” protocol. We then define a proof of quantum knowledge, which guarantees that a prover is effectively in possession of a quantum witness in an interactive proof and show that our zero-knowledge proof system satisfies this definition. Finally, we show that our proof system can be used to establish that QMA has a quantum noninteractive zero-knowledge proof system in the secret parameter setting.
      Citation: SIAM Journal on Computing
      PubDate: 2022-08-31T07:00:00Z
      DOI: 10.1137/21M140729X
      Issue No: Vol. 51, No. 4 (2022)
       
  • Constant Depth Formula and Partial Function Versions of MCSP Are Hard

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      Authors: Rahul Ilango
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Attempts to prove the intractability of the Minimum Circuit Size Problem ($\mathsf{MCSP}$) date as far back as the 1950s and are well motivated by connections to cryptography, learning theory, and average-case complexity. In this work, we make progress, on two fronts, towards showing $\mathsf{MCSP}$ is intractable under worst-case assumptions. While Masek showed in the late 1970s that the version of $\mathsf{MCSP}$ for $\mathsf{DNF}$ formulas is $\mathsf{NP}$-hard, extending this result to the case of depth-3 AND/OR formulas was open. We show that determining the minimum size of a depth-$d$ formula computing a given Boolean function is $\mathsf{NP}$-hard under quasipolynomial-time randomized reductions for all constant $d \geq 2$. Our approach is based on a method to “lift” depth-$d$ formula lower bounds to depth-$(d+1)$. This method also implies the existence of a function with a $2^{\Omega_d(n)}$ additive gap between its depth-$d$ and depth-$(d+1)$ formula complexity. We also make progress in the case of general, unrestricted circuits. We show that the version of $\mathsf{MCSP}$ where the input is a partial function (represented by a string in $\{0,1,\star\}^*$) is not in $\mathsf{P}$ under the Exponential Time Hypothesis (ETH). Intriguingly, we formulate a notion of lower bound statements being $(\mathsf{P/poly})$-recognizable that is closely related to Razborov and Rudich's definition of being $(\mathsf{P/poly})$-constructive. We show that unless there are subexponential-sized circuits computing $\mathsf{SAT}$, the collection of lower bound statements used to prove the correctness of our reductions cannot be $(\mathsf{P/poly})$-recognizable.
      Citation: SIAM Journal on Computing
      PubDate: 2022-08-31T07:00:00Z
      DOI: 10.1137/20M1383562
       
  • Random Walks and Forbidden Minors II: A
           $\mathrm{poly}(d\varepsilon^{-1})$-Query Tester for Minor-Closed
           Properties of Bounded-Degree Graphs

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      Authors: Akash Kumar, C. Seshadhri, Andrew M. Stolman
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Let $G$ be a graph with $n$ vertices and maximum degree $d$. Fix some minor-closed property $\mathcal{P}$ (such as planarity). We say that $G$ is $\varepsilon$-far from $\mathcal{P}$ if one has to remove $\varepsilon dn$ edges to make it have $\mathcal{P}$. The problem of property testing $\mathcal{P}$ was introduced in the seminal work of Benjamini, Schramm, and Shapira (Symposium on the Theory of Computing 2008) that gave a tester with query complexity triply exponential in $\varepsilon^{-1}$. Levi and Ron [ACM Trans. Algorithms, 11 (2005), 24 2015] have given the best tester to date, with a quasi-polynomial (in $\varepsilon^{-1}$) query complexity. It remained an open problem to show whether there is a property tester whose query complexity is $\mathrm{poly}(d\varepsilon^{-1})$, even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity $d\cdot \mathrm{poly}(\varepsilon^{-1})$. The previous line of work on (independent of $n$, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (Foundations of Computer Science 2018) analyzing random walk algorithms that find forbidden minors.
      Citation: SIAM Journal on Computing
      PubDate: 2022-08-16T07:00:00Z
      DOI: 10.1137/20M1312824
       
  • $O(\log^2{k}/\log\log{k})$-Approximation Algorithm for Directed Steiner
           Tree: A Tight Quasi-Polynomial Time Algorithm

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      Authors: Fabrizio Grandoni, Bundit Laekhanukit, Shi Li
      Abstract: SIAM Journal on Computing, Ahead of Print.
      In the directed Steiner tree (DST) problem, we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost subgraph that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial time, i.e., in time $n^{{poly}\log (k)}$. By assuming the projection game conjecture and ${NP}\not\subseteq{\bigcap}_{0
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-28T07:00:00Z
      DOI: 10.1137/20M1312988
       
  • Correlation Decay and Partition Function Zeros: Algorithms and Phase
           Transitions

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      Authors: Jingcheng Liu, Alistair Sinclair, Piyush Srivastava
      Abstract: SIAM Journal on Computing, Ahead of Print.
      We explore connections between the phenomenon of correlation decay (more precisely, strong spatial mixing) and the location of Lee--Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $\Delta$-regular tree) implies that strong spatial mixing on all graphs of maximum degree $\Delta$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard-core lattice gas. It also allows us to prove new results on the location of Lee--Yang zeros of the antiferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the antiferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function, significantly sharpening previous results of Sokal and others. This new extension is also of independent algorithmic interest: it allows us to give the first polynomial time deterministic approximation algorithm (a fully polynomial time approximation scheme (FPTAS)) for counting the number of $q$-colorings of a graph of maximum degree $\Delta$ provided only that $q\ge 2\Delta$, a question that has been studied intensively. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. In the case when the graph is also triangle-free, we show that our algorithm applies under the weaker condition $q \geq \alpha\Delta + \beta$, where $\alpha \approx 1.764$ and $\beta = \beta(\alpha)$ are absolute constants.
      Citation: SIAM Journal on Computing
      PubDate: 2022-07-26T07:00:00Z
      DOI: 10.1137/20M1317384
       
  • Decodable Quantum LDPC Codes beyond the $\sqrt{n}$ Distance Barrier Using
           High-Dimensional Expanders

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      Authors: Shai Evra, Tali Kaufman, Gilles Zémor
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich--Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.
      Citation: SIAM Journal on Computing
      PubDate: 2022-06-14T07:00:00Z
      DOI: 10.1137/20M1383689
       
  • An $O(\log \log m)$ Prophet Inequality for Subadditive Combinatorial
           Auctions

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      Authors: Paul Dütting, Thomas Kesselheim, Brendan Lucier
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
      Citation: SIAM Journal on Computing
      PubDate: 2022-05-19T07:00:00Z
      DOI: 10.1137/20M1382799
       
  • A Parameterized Approximation Scheme for Min $k$-Cut

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      Authors: Daniel Lokshtanov, Saket Saurabh, Vaishali Surianarayanan
      Abstract: SIAM Journal on Computing, Ahead of Print.
      In the Min $k$-Cut problem, the input consists of an edge weighted graph $G$ and an integer $k$, and the task is to partition the vertex set into $k$ nonempty sets, such that the total weight of the edges with endpoints in different parts is minimized. When $k$ is part of the input, the problem is NP-complete and hard to approximate within any factor less than 2. Recently, the problem has received significant attention from the perspective of parameterized approximation. Gupta, Lee, and Li [Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, A. Czumaj, ed., SIAM, Philadelphia, 2018, pp. 2821--2837] initiated the study of FPT-approximation for the Min $k$-Cut problem and gave a 1.9997-approximation algorithm running in time $2^{\mathcal{O}(k^6)}n^{\mathcal{O}(1)}$. Later, the same set of authors [Proceedings of the 59th IEEE Annual Symposium on Foundations of Computer Science, M. Thorup, ed., 2018, pp. 113--123] designed a $(1 +\epsilon)$-approximation algorithm that runs in time $(k/\epsilon)^{\mathcal{O}(k)}n^{k+\mathcal{O}(1)}$ and a 1.81-approximation algorithm running in time $2^{\mathcal{O}(k^2)}n^{\mathcal{O}(1)}$. More, recently, Kawarabayashi and Lin [Proceedings of the 31st ACM-SIAM Symposium on Discrete Algorithms, S. Chawla, ed., SIAM, Philadelphia, 2020, pp. 990--999] gave a $(5/3 + \epsilon)$-approximation for Min $k$-Cut running in time $2^{\mathcal{O}(k^2 \log k)}n^{\mathcal{O}(1)}$. In this paper, we give a parameterized approximation algorithm with best possible approximation guarantee and best possible running time dependence on said guarantee (up to the exponential time hypothesis and constants in the exponent). In particular, for every $\epsilon> 0$, the algorithm obtains a $(1 +\epsilon)$-approximate solution in time $(k/\epsilon)^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$. The main ingredients of our algorithm are a simple sparsification procedure, a new polynomial time algorithm for decomposing a graph into highly connected parts, and a new exact algorithm with running time $s^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ on unweighted (multi-) graphs. Here, $s$ denotes the number of edges in a minimum $k$-cut. The latter two are of independent interest.
      Citation: SIAM Journal on Computing
      PubDate: 2022-04-25T07:00:00Z
      DOI: 10.1137/20M1383197
       
  • Unit Capacity Maxflow in Almost $m^{4/3}$ Time

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      Authors: Tarun Kathuria, Yang P. Liu, Aaron Sidford
      Abstract: SIAM Journal on Computing, Ahead of Print.
      We present an algorithm which given any $m$-edge directed graph with positive integer capacities at most $U$, vertices $a$ and $b$, and an approximation parameter $\epsilon \in (0, 1)$ computes an additive $\eps mU$-approximate $a$-$b$ maximum flow in time $m^{1+o(1)}/\sqrt{\eps}$. By applying the algorithm for $\eps = (mU)^{-2/3}$, rounding to an integral flow, and using augmenting paths, we obtain an algorithm which computes an exact $a$-$b$ maximum flow in time $m^{4/3+o(1)}U^{1/3}$ and an algorithm which given an $m$-edge bipartite graph computes an exact maximum cardinality matching in time $m^{4/3+o(1)}$.
      Citation: SIAM Journal on Computing
      PubDate: 2022-04-18T07:00:00Z
      DOI: 10.1137/20M1383525
       
  • Counting Small Induced Subgraphs Satisfying Monotone Properties

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      Authors: Marc Roth, Johannes Schmitt, Philip Wellnitz
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Given a graph property $\Phi$, the problem $\#\ensuremath{{\sc IndSub}}(\Phi)$ asks, on input of a graph $G$ and a positive integer $k$, to compute the number $\#{\ensuremath{{IndSub}({\Phi,k} \to {G})}}$ of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\ensuremath{{\sc IndSub}}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. System Sci., 81 (2015), pp. 702--716] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell, and Marx [STOC, ACM, New York, pp. 151--158] proving that a full classification into “easy” and “hard” properties is possible and some partial results on edge-monotone properties due to Meeks [Discrete Appl. Math., 198 (2016), pp. 170--194] and Dörfler et al. [MFCS, LIPIcs Leibniz Int. Proc. Inform. 138, Wadern Germany, 2019, 26], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is, subgraph-closed, properties: We show that for any nontrivial monotone property $\Phi$, the problem $\#\ensuremath{{\sc IndSub}}(\Phi)$ cannot be solved in time $f(k)\cdot V(G) ^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the exponential time hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#{\ensuremath{{W[1]}}}$-completeness result. The methods we develop for the above problem also allow us to prove a conjecture by Jerrum and Meeks [ACM Trans. Comput. Theory, 7 (2015), 11; Combinatorica 37 (2017), pp. 965--990]: $\#\ensuremath{{\sc IndSub}}(\Phi)$ is $\#{\ensuremath{{W[1]}}}$-complete if $\Phi$ is a nontrivial graph property only depending on the number of edges of the graph.
      Citation: SIAM Journal on Computing
      PubDate: 2022-04-11T07:00:00Z
      DOI: 10.1137/20M1365624
       
  • Smoothing the Gap Between NP and ER

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      Authors: Jeff Erickson, Ivor van der Hoog, Tillmann Miltzow
      Abstract: SIAM Journal on Computing, Ahead of Print.
      We study algorithmic problems that belong to the complexity class of the existential theory of the reals ($\exists \mathbb{R}$). A problem is \ensuremath\exists \mathbbR-complete if it is as hard as the problem existential theory of the reals (ETR) and if it can be written as an ETR formula. Traditionally, these problems are studied in the real random access machine (RAM), a model of computation that assumes that the storage and comparison of real-valued numbers can be done in constant space and time, with infinite precision. The complexity class $\exists \mathbb{R}$ is often called a real RAM analogue of NP, since the problem ETR can be viewed as the real-valued variant of SAT. The real RAM assumption that we can represent and in which we can compare arbitrary irrational values in constant space and time is not very realistic. Yet this assumption is vital, since some $\exists \mathbb{R}$-complete problems have an “exponential bit phenomenon,” where there exists an input for the problem, such that the witness of the solution requires geometric coordinates which need exponential word size when represented in binary. The problems that exhibit this phenomenon are NP-hard (since ETR is NP-hard) but it is unknown if they lie in NP. NP membership is often showed by using the famous Cook--Levin theorem, which states that the existence of a polynomial-time verification algorithm for the problem witness is equivalent to NP membership. The exponential bit phenomenon prohibits a straightforward application of the Cook--Levin theorem. In this paper we first present a result which we believe to be of independent interest: we prove a real RAM analogue to the Cook--Levin theorem which shows that $\exists \mathbb{R}$ membership is equivalent to having a verification algorithm that runs in polynomial-time on a real RAM. This gives an easy proof of \ensuremath\exists \mathbbR-membership, as verification algorithms on a real RAM are much more versatile than ETR formulas. We use this result to construct a framework to study $\exists \mathbb{R}$-complete problems under smoothed analysis. We show that for a wide class of $\exists \mathbb{R}$-complete problems, its witness can be represented with logarithmic input-precision by using smoothed analysis on its real RAM verification algorithm. This shows in a formal way that the boundary between NP and $\exists \mathbb{R}$ (formed by inputs whose solution witness needs high input-precision) consists of contrived input. We apply our framework to well-studied $\exists \mathbb{R}$-complete recognition problems which have the exponential bit phenomenon such as the recognition of realizable order types or the Steinitz problem in fixed dimension. Interestingly our techniques also generalize to problems with a natural notion of resource augmentation (geometric packing, the art gallery problem).
      Citation: SIAM Journal on Computing
      PubDate: 2022-04-07T07:00:00Z
      DOI: 10.1137/20M1385287
       
  • Truly Optimal Euclidean Spanners

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      Authors: Hung Le, Shay Solomon
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 1980s and early 1990s state that for any $d$-dimensional $n$-point Euclidean space, there exists a $(1+\epsilon)$-spanner with $n \cdot O(\epsilon^{-d+1})$ edges and lightness (normalized weight) $O(\epsilon^{-2d})$. Surprisingly, the fundamental question of whether or not these dependencies on $\epsilon$ and $d$ for small $d$ can be improved has remained elusive, even for $d = 2$. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus $\epsilon$ is tiny). In the most extreme case $\epsilon$ is inverse polynomial in $n$, and then one could potentially improve the size and lightness bounds by factors that are polynomial in $n$. The state-of-the-art bounds $n \cdot O(\epsilon^{-d+1})$ and $O(\epsilon^{-2d})$ on the size and lightness of spanners are realized by the greedy spanner. In 2016, in a preliminary version, Filtser and Solomon [SIAM J. Comput., 49 (2020), pp. 429--447] proved that, in low-dimensional spaces, the greedy spanner is “near-optimal''; informally, their result states that the greedy spanner for dimension $d$ is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is twofold: (1) We resolve these longstanding questions by nailing down the dependencies on $\epsilon$ and $d$ and showing that the greedy spanner is truly optimal. Specifically, for any $d= O(1), \epsilon = \Omega({n}^{-\frac{1}{d-1}})$, (a) we show that there are $n$-point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have $n \cdot \Omega(\epsilon^{-d+1})$ edges, implying that the greedy (and other) spanners achieve the optimal size; (b) we show that there are $n$-point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon^{-d})$, and then improve the upper bound on the lightness of the greedy spanner from $O(\epsilon^{-2d})$ to $O(\epsilon^{-d} \log(\epsilon^{-1}))$. (The lightness upper and lower bounds match up to a lower-order term.) (2) We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners in $\mathbb{R}^2$ is tight as well (up to a lower-order term).
      Citation: SIAM Journal on Computing
      PubDate: 2022-03-31T07:00:00Z
      DOI: 10.1137/20M1317906
       
  • Performance of Johnson--Lindenstrauss Transform for $k$-Means and
           $k$-Medians Clustering

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      Authors: Konstantin Makarychev, Yury Makarychev, Ilya Razenshteyn
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Consider an instance of Euclidean $k$-means or $k$-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\varepsilon)$ under a projection onto a random $O(\log(k / \varepsilon) / \varepsilon^2)$-dimensional subspace. Further, the cost of every clustering is preserved within $(1+\varepsilon)$. More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean $k$-clustering with the distances raised to the $p$th power for any constant $p$. For $k$-means, our result resolves an open problem posed by Cohen et al. [STOC 2015, ACM, New York, 2015, pp. 163--172] for $k$-medians, it answers a question raised by Kannan.
      Citation: SIAM Journal on Computing
      PubDate: 2022-03-14T07:00:00Z
      DOI: 10.1137/20M1330701
       
  • Polynomial Data Structure Lower Bounds in the Group Model

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      Authors: Alexander Golovnev, Gleb Posobin, Oded Regev, Omri Weinstein
      Abstract: SIAM Journal on Computing, Ahead of Print.
      Proving superlogarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early '80s. We prove a polynomial ($n^{\Omega(1)}$) lower bound for an explicit range counting problem of $n^3$ convex polygons in $\mathbb{R}^2$ (each with $n^{\tilde{O}(1)}$ facets/semialgebraic complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine approximation, pseudorandomness, and compressed sensing---in particular, on the existence and partial derandomization of optimal binary compressed sensing matrices in the polynomial sparsity regime ($k = n^{1-\delta}$). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.
      Citation: SIAM Journal on Computing
      PubDate: 2022-03-14T07:00:00Z
      DOI: 10.1137/20M1381988
       
  • Efficient Construction of Rigid Matrices Using an NP Oracle

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      Authors: Josh Alman, Lijie Chen
      Abstract: SIAM Journal on Computing, Ahead of Print.
      For a matrix $H$ over a field $\mathbb{F}$, its rank-$r$ rigidity, denoted by $\mathscr{R}_{H}(r)$, is the minimum Hamming distance from $H$ to a matrix of rank at most $r$ over $\mathbb{F}$. A central open challenge in complexity theory is to give explicit constructions of rigid matrices for a variety of parameter settings. In this work, building on Williams' seminal connection between circuit-analysis algorithms and lower bounds [J. ACM, 61 (2014), 2], we give a construction of rigid matrices in $P}^{NP}}$. Letting $q = p^r$ be a prime power, we show that there is an absolute constant $\delta>0$ such that, for all constants $\varepsilon>0$, there is a $P}^{NP}}$ machine $M$ such that, for infinitely many $N$'s, $M(1^N)$ outputs a matrix $H_N \in \{0,1\}^{N \times N}$ with $\mathscr{R}_{H_N}(2^{(\log N)^{1/4 - \varepsilon}}) \ge \delta \cdot N^2$ over $\mathbb{F}_q$. Using known connections between matrix rigidity and other topics in complexity theory, we derive several consequences of our constructions, including that there is a function $f \in TIME}[2^{(\log n)^{\omega(1)}}]^{NP}}$ such that $f \notin PH}^{cc}}$. Previously, it was open whether $E}^{NP}} \subset PH}^{cc}}$. For all $\varepsilon>0$, there is a $P}^{NP}}$ machine $M$ such that, for infinitely many $N$'s, $M(1^N)$ outputs an $N \times N$ matrix $H_N \in \{0,1\}^{N \times N}$ whose linear transformation requires depth-2 $\mathbb{F}_q$-linear circuits of size $\Omega(N \cdot 2^{(\log N)^{1/4 - \varepsilon}})$. The previous best lower bound for an explicit family of $N \times N$ matrices over $\mathbb{F}_q$ was only $\Omega(N \log^2 N / (\log\log N)^2)$, for asymptotically good error correcting codes.
      Citation: SIAM Journal on Computing
      PubDate: 2022-03-07T08:00:00Z
      DOI: 10.1137/20M1322297
       
  • The Complexity of Necklace Splitting, Consensus-Halving, and Discrete Ham
           Sandwich

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      Authors: Aris Filos-Ratsikas, Paul W. Goldberg
      Abstract: SIAM Journal on Computing, Ahead of Print.
      We resolve the computational complexity of three problems known as Necklace Splitting, Consensus-Halving, and Discrete Ham sandwich, showing that they are PPA-complete. For NECKLACE SPLITTING, this result is specific to the important special case in which two thieves share the necklace. These are the first PPA-completeness results for problems whose definition does not contain an explicit circuit, thus settling the status of PPA as a class that captures the complexity of such “natural' problems.
      Citation: SIAM Journal on Computing
      PubDate: 2022-02-28T08:00:00Z
      DOI: 10.1137/20M1312678
       
  • Weak Zero-Knowledge beyond the Black-Box Barrier

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      Authors: Nir Bitansky, Dakshita Khurana, Omer Paneth
      Abstract: SIAM Journal on Computing, Ahead of Print.
      The round complexity of zero-knowledge protocols is a long-standing open question and is yet to be settled under standard assumptions. So far, the question has appeared equally challenging for relaxations such as weak zero-knowledge and witness hiding. Like full-fledged zero- knowledge, protocols satisfying these relaxed notions under standard assumptions have at least four messages. The difficulty in improving round complexity stems from a fundamental barrier: none of these notions can be achieved in three messages via reductions (or simulators) that treat the verifier as a black box. We introduce a new non-black-box technique and use it to obtain the first protocols that cross this barrier under standard assumptions. Our main results are (1) weak zero-knowledge for NP in two messages, assuming quasi-polynomially secure fully homomorphic encryption and other standard primitives (known from quasi-polynomial hardness of learning with errors) as well as subexponentially secure one-way functions; and (2) weak zero-knowledge for NP in three messages under standard polynomial assumptions (following, for example, from fully homomorphic encryption and factoring). We also give, under polynomial assumptions, a two-message witness-hiding protocol for any language L in NP that has a witness encryption scheme. This protocol is also publicly verifiable. Our technique is based on a new homomorphic trapdoor paradigm, which can be seen as a non-black-box analogue of the classic Feige--Lapidot--Shamir trapdoor paradigm.
      Citation: SIAM Journal on Computing
      PubDate: 2022-01-27T08:00:00Z
      DOI: 10.1137/20M1319565
       
 
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