Subjects -> MATHEMATICS (Total: 1118 journals)
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    - PROBABILITIES AND MATH STATISTICS (113 journals)

MATHEMATICS (819 journals)                  1 2 3 4 5 | Last

Showing 1 - 200 of 538 Journals sorted alphabetically
Abakós     Open Access   (Followers: 5)
Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg     Hybrid Journal   (Followers: 3)
Accounting Perspectives     Full-text available via subscription   (Followers: 9)
ACM Transactions on Algorithms (TALG)     Hybrid Journal   (Followers: 17)
ACM Transactions on Computational Logic (TOCL)     Hybrid Journal   (Followers: 5)
ACM Transactions on Mathematical Software (TOMS)     Hybrid Journal   (Followers: 9)
ACS Applied Materials & Interfaces     Hybrid Journal   (Followers: 44)
Acta Applicandae Mathematicae     Hybrid Journal   (Followers: 2)
Acta Mathematica     Hybrid Journal   (Followers: 11)
Acta Mathematica Hungarica     Hybrid Journal   (Followers: 2)
Acta Mathematica Scientia     Full-text available via subscription   (Followers: 5)
Acta Mathematica Sinica, English Series     Hybrid Journal   (Followers: 6)
Acta Mathematica Vietnamica     Hybrid Journal  
Acta Mathematicae Applicatae Sinica, English Series     Hybrid Journal  
Advanced Science Letters     Full-text available via subscription   (Followers: 13)
Advances in Applied Clifford Algebras     Hybrid Journal   (Followers: 6)
Advances in Calculus of Variations     Hybrid Journal   (Followers: 7)
Advances in Catalysis     Full-text available via subscription   (Followers: 8)
Advances in Complex Systems     Hybrid Journal   (Followers: 12)
Advances in Computational Mathematics     Hybrid Journal   (Followers: 23)
Advances in Decision Sciences     Open Access   (Followers: 4)
Advances in Difference Equations     Open Access   (Followers: 5)
Advances in Fixed Point Theory     Open Access   (Followers: 9)
Advances in Geosciences (ADGEO)     Open Access   (Followers: 22)
Advances in Linear Algebra & Matrix Theory     Open Access   (Followers: 10)
Advances in Materials Science     Open Access   (Followers: 22)
Advances in Mathematical Physics     Open Access   (Followers: 10)
Advances in Mathematics     Full-text available via subscription   (Followers: 22)
Advances in Numerical Analysis     Open Access   (Followers: 8)
Advances in Operations Research     Open Access   (Followers: 14)
Advances in Operator Theory     Hybrid Journal   (Followers: 4)
Advances in Porous Media     Full-text available via subscription   (Followers: 6)
Advances in Pure and Applied Mathematics     Hybrid Journal   (Followers: 12)
Advances in Pure Mathematics     Open Access   (Followers: 12)
Advances in Science and Research (ASR)     Open Access   (Followers: 8)
Aequationes Mathematicae     Hybrid Journal   (Followers: 2)
African Journal of Educational Studies in Mathematics and Sciences     Full-text available via subscription   (Followers: 12)
African Journal of Mathematics and Computer Science Research     Open Access   (Followers: 7)
Afrika Matematika     Hybrid Journal   (Followers: 3)
Air, Soil & Water Research     Open Access   (Followers: 13)
AKSIOMA Journal of Mathematics Education     Open Access   (Followers: 4)
AKSIOMATIK : Jurnal Penelitian Pendidikan dan Pembelajaran Matematika     Open Access   (Followers: 1)
Al-Jabar : Jurnal Pendidikan Matematika     Open Access   (Followers: 1)
Al-Qadisiyah Journal for Computer Science and Mathematics     Open Access   (Followers: 1)
AL-Rafidain Journal of Computer Sciences and Mathematics     Open Access   (Followers: 6)
Algebra and Logic     Hybrid Journal   (Followers: 8)
Algebra Colloquium     Hybrid Journal   (Followers: 4)
Algebra Universalis     Hybrid Journal   (Followers: 2)
Algorithmic Operations Research     Open Access   (Followers: 5)
Algorithms     Open Access   (Followers: 14)
Algorithms Research     Open Access   (Followers: 2)
American Journal of Computational and Applied Mathematics     Open Access   (Followers: 10)
American Journal of Mathematical Analysis     Open Access   (Followers: 2)
American Journal of Mathematical and Management Sciences     Hybrid Journal  
American Journal of Mathematics     Full-text available via subscription   (Followers: 9)
American Journal of Operations Research     Open Access   (Followers: 8)
American Mathematical Monthly     Full-text available via subscription   (Followers: 7)
An International Journal of Optimization and Control: Theories & Applications     Open Access   (Followers: 13)
Analele Universitatii Ovidius Constanta - Seria Matematica     Open Access  
Analysis and Applications     Hybrid Journal   (Followers: 2)
Analysis and Mathematical Physics     Hybrid Journal   (Followers: 10)
Analysis. International mathematical journal of analysis and its applications     Hybrid Journal   (Followers: 7)
Anargya : Jurnal Ilmiah Pendidikan Matematika     Open Access   (Followers: 8)
Annales Mathematicae Silesianae     Open Access   (Followers: 2)
Annales mathématiques du Québec     Hybrid Journal   (Followers: 4)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica     Open Access   (Followers: 1)
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica     Open Access  
Annali di Matematica Pura ed Applicata     Hybrid Journal   (Followers: 1)
Annals of Combinatorics     Hybrid Journal   (Followers: 3)
Annals of Data Science     Hybrid Journal   (Followers: 17)
Annals of Discrete Mathematics     Full-text available via subscription   (Followers: 8)
Annals of Functional Analysis     Hybrid Journal   (Followers: 5)
Annals of Mathematics     Full-text available via subscription   (Followers: 4)
Annals of Mathematics and Artificial Intelligence     Hybrid Journal   (Followers: 16)
Annals of PDE     Hybrid Journal  
Annals of Pure and Applied Logic     Open Access   (Followers: 6)
Annals of the Alexandru Ioan Cuza University - Mathematics     Open Access  
Annals of the Institute of Statistical Mathematics     Hybrid Journal   (Followers: 1)
Annals of West University of Timisoara - Mathematics     Open Access   (Followers: 1)
Annals of West University of Timisoara - Mathematics and Computer Science     Open Access   (Followers: 2)
Annuaire du Collège de France     Open Access   (Followers: 6)
ANZIAM Journal     Open Access   (Followers: 2)
Applicable Algebra in Engineering, Communication and Computing     Hybrid Journal   (Followers: 3)
Applications of Mathematics     Hybrid Journal   (Followers: 3)
Applied Categorical Structures     Hybrid Journal   (Followers: 4)
Applied Computational Intelligence and Soft Computing     Open Access   (Followers: 16)
Applied Mathematics     Open Access   (Followers: 12)
Applied Mathematics     Open Access   (Followers: 7)
Applied Mathematics & Optimization     Hybrid Journal   (Followers: 13)
Applied Mathematics - A Journal of Chinese Universities     Hybrid Journal   (Followers: 2)
Applied Mathematics and Nonlinear Sciences     Open Access   (Followers: 1)
Applied Mathematics Letters     Full-text available via subscription   (Followers: 3)
Applied Mathematics Research eXpress     Hybrid Journal   (Followers: 2)
Applied Network Science     Open Access   (Followers: 3)
Applied Numerical Mathematics     Hybrid Journal   (Followers: 6)
Applied Spatial Analysis and Policy     Hybrid Journal   (Followers: 6)
Arab Journal of Mathematical Sciences     Open Access   (Followers: 4)
Arabian Journal of Mathematics     Open Access   (Followers: 2)
Archive for Mathematical Logic     Hybrid Journal   (Followers: 4)
Archive of Applied Mechanics     Hybrid Journal   (Followers: 6)
Archive of Numerical Software     Open Access  
Archives of Computational Methods in Engineering     Hybrid Journal   (Followers: 6)
Arkiv för Matematik     Hybrid Journal   (Followers: 1)
Armenian Journal of Mathematics     Open Access   (Followers: 1)
Arnold Mathematical Journal     Hybrid Journal   (Followers: 1)
Artificial Satellites     Open Access   (Followers: 24)
Asia-Pacific Journal of Operational Research     Hybrid Journal   (Followers: 3)
Asian Journal of Algebra     Open Access   (Followers: 1)
Asian Research Journal of Mathematics     Open Access  
Asian-European Journal of Mathematics     Hybrid Journal   (Followers: 4)
Australian Mathematics Teacher, The     Full-text available via subscription   (Followers: 7)
Australian Primary Mathematics Classroom     Full-text available via subscription   (Followers: 7)
Australian Senior Mathematics Journal     Full-text available via subscription   (Followers: 2)
Automatic Documentation and Mathematical Linguistics     Hybrid Journal   (Followers: 5)
Axioms     Open Access   (Followers: 1)
Baltic International Yearbook of Cognition, Logic and Communication     Open Access   (Followers: 2)
Banach Journal of Mathematical Analysis     Hybrid Journal   (Followers: 1)
Basin Research     Hybrid Journal   (Followers: 6)
BIBECHANA     Open Access   (Followers: 2)
Biomath     Open Access  
BIT Numerical Mathematics     Hybrid Journal   (Followers: 1)
Boletim Cearense de Educação e História da Matemática     Open Access  
Boletín de la Sociedad Matemática Mexicana     Hybrid Journal  
Bollettino dell'Unione Matematica Italiana     Full-text available via subscription   (Followers: 3)
British Journal for the History of Mathematics     Hybrid Journal  
British Journal of Mathematical and Statistical Psychology     Full-text available via subscription   (Followers: 19)
Bruno Pini Mathematical Analysis Seminar     Open Access  
Buletinul Academiei de Stiinte a Republicii Moldova. Matematica     Open Access   (Followers: 15)
Bulletin des Sciences Mathamatiques     Full-text available via subscription   (Followers: 4)
Bulletin of Dnipropetrovsk University. Series : Communications in Mathematical Modeling and Differential Equations Theory     Open Access   (Followers: 3)
Bulletin of Mathematical Sciences     Open Access   (Followers: 1)
Bulletin of Symbolic Logic     Full-text available via subscription   (Followers: 3)
Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics     Open Access   (Followers: 1)
Bulletin of the Australian Mathematical Society     Full-text available via subscription   (Followers: 2)
Bulletin of the Brazilian Mathematical Society, New Series     Hybrid Journal  
Bulletin of the Iranian Mathematical Society     Hybrid Journal  
Bulletin of the London Mathematical Society     Hybrid Journal   (Followers: 3)
Bulletin of the Malaysian Mathematical Sciences Society     Hybrid Journal  
Cadernos do IME : Série Matemática     Open Access   (Followers: 2)
Calculus of Variations and Partial Differential Equations     Hybrid Journal  
Canadian Journal of Mathematics / Journal canadien de mathématiques     Hybrid Journal  
Canadian Journal of Science, Mathematics and Technology Education     Hybrid Journal   (Followers: 23)
Canadian Mathematical Bulletin     Hybrid Journal  
Carpathian Mathematical Publications     Open Access   (Followers: 1)
Catalysis in Industry     Hybrid Journal   (Followers: 1)
CEAS Space Journal     Hybrid Journal   (Followers: 6)
CHANCE     Hybrid Journal   (Followers: 5)
Chaos, Solitons & Fractals     Hybrid Journal   (Followers: 3)
Chaos, Solitons & Fractals : X     Open Access   (Followers: 1)
ChemSusChem     Hybrid Journal   (Followers: 8)
Chinese Annals of Mathematics, Series B     Hybrid Journal  
Chinese Journal of Catalysis     Full-text available via subscription   (Followers: 3)
Chinese Journal of Mathematics     Open Access  
Ciencia     Open Access   (Followers: 1)
CODEE Journal     Open Access   (Followers: 2)
Cogent Mathematics     Open Access   (Followers: 2)
Cognitive Computation     Hybrid Journal   (Followers: 3)
Collectanea Mathematica     Hybrid Journal  
College Mathematics Journal     Hybrid Journal   (Followers: 4)
COMBINATORICA     Hybrid Journal  
Combinatorics, Probability and Computing     Hybrid Journal   (Followers: 4)
Combustion Theory and Modelling     Hybrid Journal   (Followers: 19)
Commentarii Mathematici Helvetici     Hybrid Journal  
Communications in Advanced Mathematical Sciences     Open Access  
Communications in Combinatorics and Optimization     Open Access  
Communications in Contemporary Mathematics     Hybrid Journal  
Communications in Mathematical Physics     Hybrid Journal   (Followers: 4)
Communications On Pure & Applied Mathematics     Hybrid Journal   (Followers: 5)
Complex Analysis and its Synergies     Open Access   (Followers: 3)
Complex Variables and Elliptic Equations: An International Journal     Hybrid Journal  
Composite Materials Series     Full-text available via subscription   (Followers: 12)
Compositio Mathematica     Full-text available via subscription  
Comptes Rendus : Mathematique     Open Access  
Computational and Applied Mathematics     Hybrid Journal   (Followers: 4)
Computational and Mathematical Methods     Hybrid Journal   (Followers: 1)
Computational and Mathematical Methods in Medicine     Open Access   (Followers: 3)
Computational and Mathematical Organization Theory     Hybrid Journal   (Followers: 1)
Computational Complexity     Hybrid Journal   (Followers: 4)
Computational Mathematics and Modeling     Hybrid Journal   (Followers: 9)
Computational Mechanics     Hybrid Journal   (Followers: 11)
Computational Methods and Function Theory     Hybrid Journal  
Computational Optimization and Applications     Hybrid Journal   (Followers: 11)
Computers & Mathematics with Applications     Full-text available via subscription   (Followers: 11)
Concrete Operators     Open Access   (Followers: 4)
Confluentes Mathematici     Hybrid Journal  
Constructive Mathematical Analysis     Open Access   (Followers: 1)
Contemporary Mathematics and Statistics     Open Access   (Followers: 3)
Contributions to Discrete Mathematics     Open Access   (Followers: 1)
Contributions to Game Theory and Management     Open Access  
COSMOS     Hybrid Journal   (Followers: 1)
Cross Section     Full-text available via subscription   (Followers: 1)
Cryptography and Communications     Hybrid Journal   (Followers: 14)
Cuadernos de Investigación y Formación en Educación Matemática     Open Access  
Cubo. A Mathematical Journal     Open Access  
Current Research in Biostatistics     Open Access   (Followers: 8)
Czechoslovak Mathematical Journal     Hybrid Journal   (Followers: 1)
Daya Matematis : Jurnal Inovasi Pendidikan Matematika     Open Access   (Followers: 1)
Demographic Research     Open Access   (Followers: 16)
Design Journal : An International Journal for All Aspects of Design     Hybrid Journal   (Followers: 35)
Desimal : Jurnal Matematika     Open Access   (Followers: 3)

        1 2 3 4 5 | Last

Similar Journals
Journal Cover
Computational Complexity
Journal Prestige (SJR): 0.381
Citation Impact (citeScore): 1
Number of Followers: 4  
 
  Hybrid Journal Hybrid journal (It can contain Open Access articles)
ISSN (Print) 1420-8954 - ISSN (Online) 1016-3328
Published by Springer-Verlag Homepage  [2654 journals]
  • Lower Bounds for Arithmetic Circuits via the Hankel Matrix

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      Abstract: Abstract We study the complexity of representing polynomials by arithmetic circuits in both the commutative and the non-commutative settings. Our approach goes through a precise understanding of the more restricted setting where multiplication is not associative, meaning that we distinguish (xy)z from x(yz). Our first and main conceptual result is a characterization result: We show that the size of the smallest circuit computing a given non-associative polynomial is exactly the rank of a matrix constructed from the polynomial and called the Hankel matrix. This result applies to the class of all circuits in both commutative and non-commutative settings, and can be seen as an extension of the seminal result of Nisan giving a similar characterization for non-commutative algebraic branching programs. The study of the Hankel matrix provides a unifying approach for proving lower bounds for polynomials in the (classical) associative setting. Our key technical contribution is to provide generic lower bound theorems based on analyzing and decomposing the Hankel matrix. We obtain significant improvements on lower bounds for circuits with many parse trees, in both (associative) commutative and non-commutative settings, as well as alternative proofs of recent results proving superpolynomial and exponential lower bounds for different classes of circuits as corollaries of our characterization and decomposition results.
      PubDate: 2021-10-08
      DOI: 10.1007/s00037-021-00214-1
       
  • Near-Optimal Lower Bounds on Regular Resolution Refutations of Tseitin
           Formulas for All Constant-Degree Graphs

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      Abstract: Abstract This paper is motivated by seeking the exact complexity of resolution refutation of Tseitin formulas. We prove that the size of any regular resolution refutation of a Tseitin formula \( {\rm T}(G, c)\) based on a connected graph \({G} =(V, E)\) is at least \(2^{\Omega({\rm tw}(G)/ \log V )}\) , where \({\rm tw}(G)\) denotes the treewidth of a graph G. For constant-degree graphs, there is a known upper bound \(2^{\mathcal{O}({\rm tw}(G))}{\rm poly}( V )\) (Alekhnovich & Razborov Comput. Compl. 2011; Galesi, Talebanfard & Torán ACM Trans. Comput. Theory 2020), so our lower bound is tight up to a logarithmic factor in the exponent. Our proof consists of two steps. First, we show that any regular resolution refutation of an unsatisfiable Tseitin formula \({\rm T}(G, c) \) of size S can be converted to a read-once branching program computing a satisfiable Tseitin formula \({\rm T}(G,c')\) of size \(S^{{\mathcal{O}}({\rm log} V )}\) and this bound is tight. Second, we give the exact characterization of the nondeterministic read-once branching program (1-NBP) complexity of satisfiable Tseitin formulas in terms of structural properties of underlying graphs. Namely, we introduce a new graph measure, the component width (compw) and show that the size of a minimal \({1\text{-}\mathrm{NBP}}\) computing a satisfiable Tseitin formula \({\rm T}(G,c')\) based on a graph \({G} = (V, E)\) equals \(2^{compw}(G)\) up to a polynomial factor. Then we show that \(\Omega({\rm tw}(G)) \le {\rm compw}(G) \le {\mathcal{O}}({\rm tw}(G){\rm log}( V ))\) and both of these bounds are tight. The lower bound improves the recent result by Glinskih & Itsykson (Theory Comput. Syst. 2021).
      PubDate: 2021-08-27
      DOI: 10.1007/s00037-021-00213-2
       
  • An Exponential Separation Between MA and AM Proofs of Proximity

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      Abstract: Abstract Interactive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work, we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as MA-proofs of Proximity (MAP), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as AM-proofs of Proximity (AMP), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both MAPs and AMPs is the total number of bits that the verifier observes—namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of MAPs and AMPs. Specifically, we exhibit an explicit and natural property \(\Pi\) that admits an AMP with complexity \(O(\log n)\) , whereas any MAP for \(\Pi\) has complexity \(\tilde{\Omega}(n^{1/4})\) , where n denotes the length of the input in bits. Our MAP lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown a \(\Omega(n^{1/6})\) QMA lower bound for the same property \(\Pi\) . Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting, we show that AMPs can only be quadratically stronger than MAPs. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.
      PubDate: 2021-08-18
      DOI: 10.1007/s00037-021-00212-3
       
  • The hardest halfspace

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      Abstract: Abstract We study the approximation of halfspaces \(h:\{0,1\}^n\to\{0,1\}\) in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the “hardest” halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of O(n) versus \(2^{-\Omega(n)}\) , between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of log n versus \(\Omega(n)\) between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the k-party number-on-the-forehead model, where we obtain an explicit separation of log n versus \(\Omega(n/4^{n})\) for communication with unbounded versus weakly unbounded error.
      PubDate: 2021-08-03
      DOI: 10.1007/s00037-021-00211-4
       
  • Nondeterministic and Randomized Boolean Hierarchies in Communication
           Complexity

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      Abstract: Abstract We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function that has an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sidederror randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.
      PubDate: 2021-07-02
      DOI: 10.1007/s00037-021-00210-5
       
  • Correction to: Smooth and Strong PCPs

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      Abstract:
      Authors would like to correct the error in their publication.
      PubDate: 2021-06-10
      DOI: 10.1007/s00037-021-00208-z
       
  • Blackbox identity testing for sum of special ROABPs and its border class

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      Abstract: Abstract We look at the problem of blackbox polynomial identity testing (PIT) for the model of read-once oblivious algebraic branching programs (ROABP), where the number of variables is logarithmic to the input size of ROABP. We restrict width of ROABP to a constant and study the more general sum-of-ROABPs model. This model is nontrivial due to the arbitrary individual-degree. We give the first poly( \(s\) )-time blackbox PIT for sum of constant-many, size- \(s\) , \(O(log s)\) -variate constant-width ROABPs. The previous best for this model was quasi-polynomial time (Gurjar et al, CCC'15; Computational Complexity'17) which is comparable to brute-force in the log-variate setting. We also show that we can work with unbounded-many such ROABPs if each ROABP computes a homogeneous polynomial (or more generally for degree-preserving sums). We also give poly-time PIT for the border. We introduce two new techniques, both of which also work for the border version of the stated models. (1) The leading-degree-part of an ROABP can be made syntactically homogeneous in the same width. (2) There is a direct reduction from PIT of sum-of-ROABPs to PIT of single ROABP (over any field). Our methods improve the time complexity for PIT of sum-of-ROABPs in the log-variate regime.
      PubDate: 2021-06-10
      DOI: 10.1007/s00037-021-00209-y
       
  • Reversible Pebble Games and the Relation Between Tree-Like and General
           Resolution Space

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      Abstract: Abstract We show a new connection between the clause space measure in tree-like resolution and the reversible pebble game on graphs. Using this connection, we provide several formula classes for which there is a logarithmic factor separation between the clause space complexity measure in tree-like and general resolution. We also provide upper bounds for tree-like resolution clause space in terms of general resolution clause and variable space. In particular, we show that for any formula F, its tree-like resolution clause space is upper bounded by space \((\pi)\) \((\log({\rm time}(\pi))\) , where \(\pi\) is any general resolution refutation of F. This holds considering as space \((\pi)\) the clause space of the refutation as well as considering its variable space. For the concrete case of Tseitin formulas, we are able to improve this bound to the optimal bound space \((\pi)\log n\) , where n is the number of vertices of the corresponding graph
      PubDate: 2021-05-01
      DOI: 10.1007/s00037-021-00206-1
       
  • Lower Bounds for Matrix Factorization

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      Abstract: Abstract We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of lower bounds for algebraic circuits which compute linear transformations, matrix rigidity and data structure lower bounds. We first show, for every constant d, a deterministic construction in time \({\rm exp}(n^{1-\Omega(1/d)})\) of a family \(\{M_n\}\) of \(n \times n\) matrices which cannot be expressed as a product \(M_n = A_1 \cdots A_d\) where the total sparsity of \(A_1,\ldots,A_d\) is less than \(n^{1+1/(2d)}\) . In other words, any depth-d linear circuit computing the linear transformation \(M_n\cdot {\bf x}\) has size at least \(n^{1+\Omega(1/d)}\) . The prior best lower bounds for this problem were barely super-linear, and were obtained by a long line of research based on the study of super-concentrators. We improve these lower bounds at the cost of a blow up in the time required to construct these matrices. Previously, however, such constructions were not known even in time \(2^{O(n)}\) with the aid of an NP oracle. We then outline an approach for proving improved lower bounds through a certain derandomization problem, and use this approach to prove asymptotically optimal quadratic lower bounds for natural special cases, which generalize many of the common matrix decompositions.
      PubDate: 2021-04-02
      DOI: 10.1007/s00037-021-00205-2
       
  • Subquadratic-Time Algorithms for Normal Bases

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      Abstract: Abstract For any finite Galois field extension K/F, with Galois group G = Gal (K/F), there exists an element \(\alpha \in \) K whose orbit \(G\cdot\alpha\) forms an F-basis of K. Such an \(\alpha\) is called a normal element, and \(G\cdot\alpha\) is a normal basis. We introduce a probabilistic algorithm for testing whether a given \(\alpha \in\) K is normal, when G is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether \(\alpha\) is normal can be reduced to deciding whether \(\sum_{g \in G} g(\alpha)g \in\) K[G] is invertible; it requires a slightly subquadratic number of operations. Once we know that \(\alpha\) is normal, we show how to perform conversions between the power basis of K/F and the normal basis with the same asymptotic cost.
      PubDate: 2021-03-02
      DOI: 10.1007/s00037-020-00204-9
       
  • Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

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      Abstract: Abstract We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.
      PubDate: 2021-02-12
      DOI: 10.1007/s00037-020-00201-y
       
  • Explicit List-Decodable Codes with Optimal Rate for Computationally
           Bounded Channels

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      Abstract: Abstract A stochastic code is a pair of encoding and decoding procedures (Enc, Dec) where \({{\rm Enc} : \{0, 1\}^{k} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{n}}\) . The code is (p, L)-list decodable against a class \(\mathcal{C}\) of “channel functions” \(C : \{0,1\}^{n} \rightarrow \{0,1\}^{n}\) if for every message \(m \in \{0,1\}^{k}\) and every channel \(C \in \mathcal{C}\) that induces at most pn errors, applying Dec on the “received word” C(Enc(m,S)) produces a list of at most L messages that contain m with high probability over the choice of uniform \(S \leftarrow \{0, 1\}^{d}\) . Note that both the channel C and the decoding algorithm Dec do not receive the random variable S, when attempting to decode. The rate of a code is \(R = k/n\) , and a code is explicit if Enc, Dec run in time poly(n). Guruswami and Smith (Journal of the ACM, 2016) showed that for every constants \(0 < p < \frac{1}{2}, \epsilon > 0\) and \(c > 1\) there exist a constant L and a Monte Carlo explicit constructions of stochastic codes with rate \(R \geq 1-H(p) - \epsilon\) that are (p, L)-list decodable for size \(n^c\) channels. Here, Monte Carlo means that the encoding and decoding need to share a public uniformly chosen \({\rm poly}(n^c)\) bit string Y, and the constructed stochastic code is (p, L)-list decodable with high probability over the choice of Y. Guruswami and Smith pose an open problem to give fully explicit (that is not Monte Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper, we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97). Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against \(O(\log n)\) -space online channels. (These are channels that have space \(O(\log n)\) and are allowed to read the input codeword in one pass.) We also resolve this open problem. Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching \(1 - H(p)\) for every \(p \leq p_{0}\) for some \(p_{0} >0\) ) for channels that are circuits of size \(2^{n^{\Omega(1/d)}}\) and depth d. Here, the running time of encoding and decoding is a polynomial that does not depend on the dept...
      PubDate: 2021-01-20
      DOI: 10.1007/s00037-020-00203-w
       
  • Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

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      Abstract: Abstract Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted \({\rm Res}({\rm lin}_R)\) , this refutation system operates with disjunctions of linear equations with Boolean variables over a ring R, to refute unsatisfiable sets of such disjunctions. Beginning in the work of Raz & Tzameret (2008), through the work of Itsykson & Sokolov (2020) which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. Garlik & Kołodziejczyk 2018; Itsykson & Sokolov 2020; Krajícek 2017; Krajícek & Oliveira 2018) made it evident that establishing lower bounds against general \({\rm Res}({\rm lin}_R)\) refutations is a challenging and interesting task since the system captures a ``minimal'' extension of resolution with counting gates for which no super-polynomial lower bounds are known to date. We provide the first super-polynomial size lower bounds against general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular, we prove that the subset-sum principle \(1+\sum\nolimits_{i=1}^{n}2^i x_i = 0\) requires refutations of exponential size over \(\mathbb{Q}\) . We use a novel lower bound technique: We show that under certain conditions every refutation of a subset-sum instance \(f=0\) must pass through a fat clause consisting of the equation \(f=\alpha\) for every \(\alpha\) in the image of f under Boolean assignments, or can be efficiently reduced to a proof containing such a clause. We then modify this approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on n variables, hence also separating tree-like from dag-like refutations over the rationals. We then turn to the finite fields regime, showing that the work of Itsykson & Sokolov (2020), where tree-like lower bounds over \(\mathbb{F}_2\) were obtained, can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (i) For every pair of distinct primes \(p,q\) , there exist CNF formulas with short tree-like refutations in \({\rm Res}({\rm lin}{\mathbb{F}_p})\) that require exponential-size tree-like \({\rm Res}({\rm lin}{\mathbb{F}_q})\) refutations; (ii) random k-CNF formulas require exponential-size tree-like \({\rm Res}({\rm lin}{\mathbb{F}_p})\) refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like \({\rm Res}({\rm lin}{\mathbb{F}})\) refutations of the pigeonhole principle, for every field \(\mathbb{F}\) .
      PubDate: 2021-01-08
      DOI: 10.1007/s00037-020-00202-x
       
  • Smooth and Strong PCPs

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      Abstract: Abstract Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: \(\circ \quad\) A PCP is strong if it rejects an alleged proof of a correct claimwith probability proportional to its distance from some correctproof of that claim. \(\circ \quad\) A PCP is smooth if each location in a proof is queried with equalprobability. We prove that all sets in \(\mathcal{NP}\) have PCPs that are both smooth andstrong, are of polynomial length and can be verified based on a constantnumber of queries. This is achieved by following the proof of thePCP theorem of Arora et al. (JACM 45(3):501–555, 1998), providing astronger analysis of the Hadamard and Reed–Muller based PCPs anda refined PCP composition theorem. In fact, we show that any set in \(\mathcal{NP}\) has a smooth strong canonical PCP of Proximity (PCPP), meaningthat there is an efficiently computable bijection of \(\mathcal{NP}\) witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of “stable” 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.
      PubDate: 2021-01-06
      DOI: 10.1007/s00037-020-00199-3
       
  • Linear Matroid Intersection is in Quasi-NC

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      Abstract: Abstract Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size \(n^{O(\log n)}\) and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log2 n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form \(A_{0} + A_{1 }x_{1} + \cdots + A_{m} x_{m}\) , for an arbitrary matrix A0 and rank-1 matrices \(A_{1}, A_{2}, \dots, A_{m}\) . This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.
      PubDate: 2020-11-19
      DOI: 10.1007/s00037-020-00200-z
       
  • The Computational Complexity of Plethysm Coefficients

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      Abstract: Abstract In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper, we show that deciding positivity of plethysm coefficients is NP-hard and that computing plethysm coefficients is #P-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.
      PubDate: 2020-11-04
      DOI: 10.1007/s00037-020-00198-4
       
  • The Robustness of LWPP and WPP, with an Application to Graph
           Reconstruction

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      Abstract: Abstract We show that the counting class LWPP remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques. The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., \(\rm PP^{Legitimate Deck} = PP\) ) if the weakened version of the Reconstruction Conjecture holds in which the number of nonisomorphic preimages is assumed merely to be polynomially bounded. This strengthens the 1992 result of Köbler, Schöning & Torán that the Legitimate Deck Problem is in LWPP if the Reconstruction Conjecture holds, and provides strengthened evidence that the Legitimate Deck Problem is not NP-hard. We additionally show on the one hand that our LWPP robustness result also holds for WPP, and also holds even when one allows both the rejection and acceptance gap-value targets to simultaneously be polynomial-sized lists; yet on the other hand, we show that for the \(\#{\rm P}\) -based analogue of LWPP the behavior much differs in that, in some relativized worlds, even two target values already yield a richer class than one value does. Despite that nonrobustness result for a \(\#{\rm P}\) -based class, we show that the \(\#{\rm P}\) -based “exact counting” class \({\rm C}_{=}{\rm P}\) remains unchanged even if one allows a polynomial number of target values for the number of accepting paths of the machine.
      PubDate: 2020-10-29
      DOI: 10.1007/s00037-020-00197-5
       
  • On $$\epsilon$$ ϵ -sensitive monotone computations

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      Abstract: Abstract We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial \(f\in \mathbb {R}[x_1,\dots , x_n]\) of degree d has an arithmetic circuit of size s then \((x_1+\dots +x_n+1)^d+\epsilon f\) has a monotone arithmetic circuit of size \(O(sd^2+n\log n)\) , for some \(\epsilon >0\) . Second, if \(f:\{0,1\}^n\rightarrow \{0,1\}\) is a Boolean function, we associate with f an explicit exponential-size matrix M(f) such that the Boolean circuit size of f is at least \(\varOmega (\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))- 2n)\) , where J is the all-ones matrix and \(\mathrm{rk}_{+}\) denotes the nonnegative rank of a matrix. In fact, the quantity \(\min _{\epsilon >0}(\mathrm{rk}_{+}(M(f)-\epsilon J))\) characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar \(\epsilon \) -sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system.
      PubDate: 2020-07-25
      DOI: 10.1007/s00037-020-00196-6
       
  • Two-closures of supersolvable permutation groups in polynomial time

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      Abstract: Abstract The 2-closure \(\overline{G}\) of a permutation group G on \(\Omega\) is defined to be the largest permutation group on \(\Omega\) , having the same orbits on \(\Omega \times \Omega\) as G. It is proved that ifG is supersolvable, then \(\overline{G}\) can be found in polynomial time in \( \Omega \) . As a by-product of our technique, it is shown that the composition factors of \(\overline{G}\) are cyclic or alternating.
      PubDate: 2020-06-24
      DOI: 10.1007/s00037-020-00195-7
       
  • Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation

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      Abstract: Abstract One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., \(\textbf{P} \not\subseteq \textbf{NC}^{1}\) ). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4):191–204, 1995) suggested to approach this problem by proving that depth complexity behaves ``as expected'' with respect to the composition of functions f ◊ g. They showed that the validity of this conjecture would imply that \(\textbf{P} \not\subseteq \textbf{NC}^{1}\) . As a way to realize this program, Edmonds et al. (Computational Complexity 10(3):210–246, 2001) suggested to study the ``multiplexor relation'' MUX. In this paper, we present two results regarding this relation: The multiplexor relation is ``complete'' for the approach of Karchmer et al. in the following sense: if we could prove (a variant of) their conjecture for the composition f ◊ MUX for every function f, then this would imply \(\textbf{P} \not\subseteq \textbf{NC}^{1}\) . A simpler proof of a lower bound for the multiplexor relation due to Edmonds et al. Our proof has the additional benefit of fitting better with the machinery used in previous works on the subject.
      PubDate: 2020-06-06
      DOI: 10.1007/s00037-020-00194-8
       
 
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