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On Blocky Ranks Of Matrices Comput. Complex. (IF 1.4) Pub Date : 2024-03-06
Abstract A matrix is blocky if it is a "blowup" of a permutation matrix. The blocky rank of a matrix M is the minimum number of blocky matrices that linearly span M. Hambardzumyan, Hatami and Hatami defined blocky rank and showed that it is connected to communication complexity and operator theory. We describe additional connections to circuit complexity and combinatorics, and we prove upper and lower
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Approximating the chromatic polynomial is as hard as computing it exactly Comput. Complex. (IF 1.4) Pub Date : 2024-01-18 Ferenc Bencs, Jeroen Huijben, Guus Regts
We show that for any non-real algebraic number q, such that \(|q-1|>1\) or \(\Re(q)>\frac{3}{2}\) it is #P-hard to compute a multiplicative (resp. additive) approximation to the absolute value (resp. argument) of the chromatic polynomial evaluated at q on planar graphs. This implies #P-hardness for all non-real algebraic q on the family of all graphs. We, moreover, prove several hardness results for
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Improving $$3N$$ Circuit Complexity Lower Bounds Comput. Complex. (IF 1.4) Pub Date : 2023-12-16 Magnus Gausdal Find , Alexander Golovnev , Edward A. Hirsch , Alexander S. Kulikov
While it can be easily shown by counting that almost all Boolean predicates of n variables have circuit size \(\Omega(2^n/n)\), we have no example of an NP function requiring even a superlinear number of gates. Moreover, only modest linear lower bounds are known. Until recently, the strongest known lower bound was \(3n - o(n)\) presented by Blum in 1984. In 2011, Demenkov and Kulikov presented a much
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On vanishing sums of roots of unity in polynomial calculus and sum-of-squares Comput. Complex. (IF 1.4) Pub Date : 2023-11-12 Ilario Bonacina, Nicola Galesi, Massimo Lauria
We introduce a novel take on sum-of-squares that is able to reason with complex numbers and still make use of polynomial inequalities. This proof system might be of independent interest since it allows to represent multivalued domains both with Boolean and Fourier encoding. We show degree and size lower bounds in this system for a natural generalization of knapsack: the vanishing sums of roots of unity
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A Lower Bound on the Complexity of Testing Grained Distributions Comput. Complex. (IF 1.4) Pub Date : 2023-10-26 Oded Goldreich , Dana Ron
For a natural number \(m\), a distribution is called \(m\)-grained, if each element appears with probability that is an integer multiple of \(1/m\). We prove that, for any constant \(c<1\), testing whether a distribution over \([\Theta(m)]\) is \(m\)-grained requires \(\Omega(m^c)\) samples, where testing a property of distributions means distinguishing between distributions that have the property
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Parallel algorithms for power circuits and the word problem of the Baumslag group Comput. Complex. (IF 1.4) Pub Date : 2023-09-13 Caroline Mattes, Armin Weiß
Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and \((x,y) \mapsto x\cdot 2^y\). The same authors applied power circuits to give a polynomial time solution to the word problem of the Baumslag group, which has a non-elementary Dehn function. In this work, we examine power
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On Time-Space Tradeoffs for Bounded-Length Collisions in Merkle-Damgård Hashing Comput. Complex. (IF 1.4) Pub Date : 2023-09-13 Ashrujit Ghoshal, Ilan Komargodski
We study the power of preprocessing adversaries in finding bounded-length collisions in the widely used Merkle-Damgård (MD) hashing in the random oracle model. Specifically, we consider adversaries with arbitrary S-bit advice about the random oracle and can make at most T queries to it. Our goal is to characterize the advantage of such adversaries in finding a B-block collision in an MD hash function
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Absolute reconstruction for sums of powers of linear forms: degree 3 and beyond Comput. Complex. (IF 1.4) Pub Date : 2023-08-03 Pascal Koiran, Subhayan Saha
We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomised algorithm for the following problem: If a homogeneous polynomial \(f \in K[x_1,..., x_n]\) (where \(K \subseteq \mathbb {C}\)) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main
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A Characterization of Functions over the Integers Computable in Polynomial Time Using Discrete Ordinary Differential Equations Comput. Complex. (IF 1.4) Pub Date : 2023-07-12 Olivier Bournez, Arnaud Durand
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several
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The Power of Natural Properties as Oracles Comput. Complex. (IF 1.4) Pub Date : 2023-07-06 Russell Impagliazzo, Valentine Kabanets, Ilya Volkovich
We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that \(\mathsf{ZPEXP}^\mathsf{MCSP}\nsubseteq \mathsf{P} / \mathsf
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Is it possible to improve Yao’s XOR lemma using reductions that exploit the efficiency of their oracle? Comput. Complex. (IF 1.4) Pub Date : 2023-05-23 Ronen Shaltiel
Yao’s XOR lemma states that for every function \(f:\{0,1\}^k \rightarrow \{0,1\}\), if f has hardness 2/3 for P/poly (meaning that for every circuit C in P/poly, \(\Pr[C(X)=f(X)] \le 2/3\) on a uniform input X), then the task of computing \(f(X_1) \oplus \ldots \oplus f(X_t)\) for sufficiently large t has hardness \(\frac{1}{2} + \epsilon\) for P/poly. Known proofs of this lemma cannot achieve \(\
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A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory Comput. Complex. (IF 1.4) Pub Date : 2023-05-17 Jin-Yi Cai, Zhiguo Fu , Kurt Girstmair, Michael Kowalczyk
Suppose \(\varphi\) and \(\psi\) are two angles satisfying \(\tan(\varphi) = 2 \tan(\psi) > 0\). We prove that under this condition \(\varphi\) and \(\psi\) cannot be both rational multiples of π. We use this number theoretic result to prove a classification of the computational complexity of spin systems on k-regular graphs with general (not necessarily symmetric) real valued edge weights. We establish
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Schur Polynomials Do Not Have Small Formulas If the Determinant does not Comput. Complex. (IF 1.4) Pub Date : 2023-05-13 Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, Srikanth Srinivasan
Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of symmetric functions, in Representation theory Stanley (1999), in Schubert calculus Ledoux & Malham (2010) as well as in Enumerative combinatorics Gasharov (1996); Stanley (1984, 1999). In recent years, they have also shown up in various
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Explicit construction of q+1 regular local Ramanujan graphs, for all prime-powers q Comput. Complex. (IF 1.4) Pub Date : 2023-04-06 Rishabh Batra, Nitin Saxena, Devansh Shringi
A constant locality function is one in which each output bit depends on just a constant number of input bits. Viola and Wigderson (2018) gave an explicit construction of bipartite degree-3 Ramanujan graphs such that each neighbor of a vertex can be computed using a constant locality function. In this work, we construct the first explicit local Ramanujan graph (bipartite) of degree q + 1, where q >
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On sets of linear forms of maximal complexity Comput. Complex. (IF 1.4) Pub Date : 2022-11-29 Michael Kaminski, Igor E. Shparlinski, Michel Waldschmidt
We present a uniform description of sets of m linear forms in n variables over the field of rational numbers whose computation requires m(n – 1) additions.
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Quantum versus Randomized Communication Complexity, with Efficient Players Comput. Complex. (IF 1.4) Pub Date : 2022-10-28 Uma Girish, Ran Raz, Avishay Tal
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity
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On Hitting-Set Generators for Polynomials that Vanish Rarely Comput. Complex. (IF 1.4) Pub Date : 2022-10-22 Dean Doron, Amnon Ta-Shma, Roei Tell
The problem of constructing pseudorandom generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct pseudorandom generators, or even hitting-set generators, for polynomials \(p:\mathbb{F}^n\rightarrow\mathbb{F}\) of degree d if we are guaranteed
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Improved Hitting Set for Orbit of ROABPs Comput. Complex. (IF 1.4) Pub Date : 2022-10-12 Vishwas Bhargava, Sumanta Ghosh
The orbit of an n-variate polynomial \(f({\rm x})\) over a field \(\mathbb{F}\) is the set \(\{f(A{\rm x} + {\rm b})\,\mid\, A\in{\rm GL}(n, \mathbb{F})\mbox{ and }{\rm b}\in \mathbb{F}^n\}\), and the orbit of a polynomial class is the union of orbits of the polynomials in it. In this paper, we give improved constructions of hitting sets for the orbit of read-once oblivious algebraic branching programs
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The Complexity of Finding Fair Independent Sets in Cycles Comput. Complex. (IF 1.4) Pub Date : 2022-10-11 Ishay Haviv
Let G be a cycle graph and let \(V_1,\ldots,V_m\) be a partition of its vertex set into m sets. An independent set S of G is said to fairly represent the partition if \(|S \cap V_i| \geq \frac{1}{2} \cdot |V_i| -1\) for all \(i \in [m]\). It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al. 2017)
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Quantum generalizations of the polynomial hierarchy with applications to QMA(2) Comput. Complex. (IF 1.4) Pub Date : 2022-09-20 Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, Justin Yirka
The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, \(\rm{QCPH}\), uses classical proofs, and the second, \(\rm{QPH}\), uses quantum
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A Lower Bound on Determinantal Complexity Comput. Complex. (IF 1.4) Pub Date : 2022-09-19 Mrinal Kumar, Ben Lee Volk
The determinantal complexity of a polynomial \(P \in \mathbb{F}[x_1, \ldots, x_n]\) over a field \(\mathbb{F}\) is the dimension of the smallest matrix M whose entries are affine functions in \(\mathbb{F}[x_1, \ldots, x_n]\) such that \(P = {\rm Det}(M)\). We prove that the determinantal complexity of the polynomial \(\sum_{i = 1}^n x_i^n\) is at least \(1.5n - 3\). For every n-variate polynomial of
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Zeros and approximations of Holant polynomials on the complex plane Comput. Complex. (IF 1.4) Pub Date : 2022-07-27 Katrin Casel, Philipp Fischbeck, Tobias Friedrich, Andreas Göbel, J. A. Gregor Lagodzinski
We present fully polynomial time approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known as polymers in statistical physics. Our method involves establishing zero-free regions for the partition functions of polymer models and using the most
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A cost-scaling algorithm for computing the degree of determinants Comput. Complex. (IF 1.4) Pub Date : 2022-07-23 Hiroshi Hirai, Motoki Ikeda
In this paper, we address computation of the degree \(\deg {\rm Det} A\) of Dieudonné determinant \({\rm Det} A\) of $$\begin{aligned} A = \sum_{k=1}^m A_k x_k t^{c_k}, \end{aligned}$$ where \(A_k\) are \(n \times n\) matrices over a field \(\mathbb{K}\), \(x_k\) are noncommutative variables, t is a variable commuting with \(x_k\), \(c_k\) are integers, and the degree is considered for t. This problem
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Disjointness through the Lens of Vapnik–Chervonenkis Dimension: Sparsity and Beyond Comput. Complex. (IF 1.4) Pub Date : 2022-07-05 Anup Bhattacharya, Sourav Chakraborty, Arijit Ghosh, Gopinath Mishra, Manaswi Paraashar
The disjointness problem—where Alice and Bob are given two subsets of \(\{1, \dots, n\}\) and they have to check if their sets intersect—is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be \(\Theta(n)\), it is also known that if the sets are assumed to be drawn from some restricted set systems
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Quadratic Lower Bounds for Algebraic Branching Programs and Formulas Comput. Complex. (IF 1.4) Pub Date : 2022-07-05 Prerona Chatterjee, Mrinal Kumar, Adrian She, Ben Lee Volk
We show that any Algebraic Branching Program (ABP) computing the polynomial \(\sum _{i = 1}^n x_i^n\) has at least \(\Omega (n^2)\) vertices. This improves upon the lower bound of \(\Omega (n\log n)\), which follows from the classical result of Strassen (1973a) and Baur & Strassen (1983), and extends the results in Kumar (2019), which showed a quadratic lower bound for homogeneous ABPs computing the
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Improved bounds on the AN-complexity of $$O(1)$$ O ( 1 ) -linear functions Comput. Complex. (IF 1.4) Pub Date : 2022-06-23 Oded Goldreich
We consider arithmetic circuits with arbitrary gates for computing Boolean functions that are represented by low-degree polynomials over GF(2). An adequate complexity measure for such circuits is the maximum between the arity of the gates and their number. This model and the corresponding complexity measure, called AN-complexity, were introduced by Goldreich and Wigderson (ECCC, TR13-043, 2013), and
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A quasi-linear irreducibility test in $$\mathbb{K}[[x]][y]$$ K [ [ x ] ] [ y ] Comput. Complex. (IF 1.4) Pub Date : 2022-06-11 Adrien Poteaux, Martin Weimann
We provide an irreducibility test in the ring \(\mathbb{K}[[x]][y]\) whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial \(F\) is square-free and \(\mathbb{K}\) is a perfect field of characteristic not dividing \(\deg(F)\). The algorithm uses the theory of approximate roots and may be seen as a generalisation of Abhyankar's irreducibility criterion
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Computing zero-dimensional tropical varieties via projections Comput. Complex. (IF 1.4) Pub Date : 2022-05-20 Paul Görlach , Yue Ren, Leon Zhang
We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast monomial transforms of triangular sets. Given a Gröbner basis, we prove that our algorithm requires only a polynomial number of arithmetic operations, and, for ideals in shape position, we show that its timings compare well against univariate factorization and backsubstitution. We conclude
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Expander-Based Cryptography Meets Natural Proofs Comput. Complex. (IF 1.4) Pub Date : 2022-03-16 Igor C. Oliveira, Rahul Santhanam, Roei Tell
We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich’s pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander’s neighbor function and/or of the local predicate. Our two key conceptual contributions are: 1. We put forward the possibility that the choice of expander matters in expander-based cryptography
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Amplification with One NP Oracle Query Comput. Complex. (IF 1.4) Pub Date : 2022-02-05 Thomas Watson
We provide a complete picture of the extent to which amplification of success probability is possible for randomized algorithms having access to one NP oracle query, in the settings of two-sided, onesided, and zero-sided error. We generalize this picture to amplifying one-query algorithms with q-query algorithms, and we show our inclusions are tight for relativizing techniques.
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The complexity of approximating the complex-valued Potts model Comput. Complex. (IF 1.4) Pub Date : 2022-02-03 Andreas Galanis, Leslie Ann Goldberg, Andrés Herrera-Poyatos
We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location
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Rank and border rank of Kronecker powers of tensors and Strassen's laser method Comput. Complex. (IF 1.4) Pub Date : 2021-12-18 Austin Conner, Fulvio Gesmundo, Joseph M. Landsberg, Emanuele Ventura
We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor \(T_{cw,q}\) is the square of its border rank for \(q > 2\) and that the border rank of its Kronecker cube is the cube of its border rank for \(q > 4\). This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility
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Correction to: Near-Optimal Lower Bounds on Regular Resolution Refutations of Tseitin Formulas for All Constant-Degree Graphs Comput. Complex. (IF 1.4) Pub Date : 2021-11-17 Dmitry Itsykson,Artur Riazanov,Danil Sagunov,Petr Smirnov
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Factorization of Polynomials Given by Arithmetic Branching Programs Comput. Complex. (IF 1.4) Pub Date : 2021-10-15 Amit Sinhababu , Thomas Thierauf
Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly \( (s^{ {\rm \log} s}) \).
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Lower Bounds for Arithmetic Circuits via the Hankel Matrix Comput. Complex. (IF 1.4) Pub Date : 2021-10-08 Nathanaël Fijalkow, Guillaume Lagarde, Pierre Ohlmann, Olivier Serre
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
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Near-Optimal Lower Bounds on Regular Resolution Refutations of Tseitin Formulas for All Constant-Degree Graphs Comput. Complex. (IF 1.4) Pub Date : 2021-08-27 Itsykson, Dmitry, Riazanov, Artur, Sagunov, Danil, Smirnov, Petr
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
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An Exponential Separation Between MA and AM Proofs of Proximity Comput. Complex. (IF 1.4) Pub Date : 2021-08-18 Gur, Tom, Liu, Yang P., Rothblum, Ron D.
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
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The hardest halfspace Comput. Complex. (IF 1.4) Pub Date : 2021-08-03 Sherstov, Alexander A.
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
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Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity Comput. Complex. (IF 1.4) Pub Date : 2021-07-02 Toniann Pitassi, Morgan Shirley, Thomas Watson
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
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Correction to: Smooth and Strong PCPs Comput. Complex. (IF 1.4) Pub Date : 2021-06-10 Orr Paradise
Authors would like to correct the error in their publication.
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Blackbox identity testing for sum of special ROABPs and its border class Comput. Complex. (IF 1.4) Pub Date : 2021-06-10 Pranav Bisht, Nitin Saxena
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
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Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space Comput. Complex. (IF 1.4) Pub Date : 2021-05-01 Jacobo Torán, Florian Wörz
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
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Lower Bounds for Matrix Factorization Comput. Complex. (IF 1.4) Pub Date : 2021-04-02 Ben Lee Volk, Mrinal Kumar
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
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Subquadratic-Time Algorithms for Normal Bases Comput. Complex. (IF 1.4) Pub Date : 2021-03-02 Mark Giesbrecht, Armin Jamshidpey, Éric Schost
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
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Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling Comput. Complex. (IF 1.4) Pub Date : 2021-02-12 Susanna F. De Rezende, Or Meir, Jakob Nordström, Robert Robere
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
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Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels Comput. Complex. (IF 1.4) Pub Date : 2021-01-20 Ronen Shaltiel, Jad Silbak
A stochastic code is a pair of encoding and decoding procedures (Enc, Dec) where $${{\rm Enc} : \{0, 1\}^{k} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{n}}$$ Enc : { 0 , 1 } k × { 0 , 1 } d → { 0 , 1 } n . The code is ( p, L )-list decodable against a class $$\mathcal{C}$$ C of “channel functions” $$C : \{0,1\}^{n} \rightarrow \{0,1\}^{n}$$ C : { 0 , 1 } n → { 0 , 1 } n if for every message $$m \in
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Resolution with Counting: Dag-Like Lower Bounds and Different Moduli Comput. Complex. (IF 1.4) Pub Date : 2021-01-08 Fedor Part, Iddo Tzameret
Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted $${\rm Res}({\rm lin}_R)$$ Res ( lin R ) , this refutation system operates with disjunctions of linear equations with Boolean variables over a ring R , to refute unsatisfiable sets of such disjunctions. Beginning in the work of Raz & Tzameret
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Smooth and Strong PCPs Comput. Complex. (IF 1.4) Pub Date : 2021-01-06 Orr Paradise
Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: $$\circ \quad$$ ∘ A PCP is strong if it rejects an alleged proof of a correct claim with
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Linear Matroid Intersection is in Quasi-NC Comput. Complex. (IF 1.4) Pub Date : 2020-11-19 Rohit Gurjar , Thomas Thierauf
Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits
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The Computational Complexity of Plethysm Coefficients Comput. Complex. (IF 1.4) Pub Date : 2020-11-04 Nick Fischer, Christian Ikenmeyer
In two papers, Burgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal
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The Robustness of LWPP and WPP, with an Application to Graph Reconstruction Comput. Complex. (IF 1.4) Pub Date : 2020-10-29 Edith Hemaspaandra, Lane A. Hemaspaandra, Holger Spakowski, Osamu Watanabe
We show that the counting class LWPP remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques. The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction)
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On $$\epsilon$$-sensitive monotone computations Comput. Complex. (IF 1.4) Pub Date : 2020-07-25 Pavel Hrubeš
We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial $$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
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Two-closures of supersolvable permutation groups in polynomial time Comput. Complex. (IF 1.4) Pub Date : 2020-06-01 Ilia Ponomarenko, Andrey Vasil’ev
The 2-closure $$\overline{G}$$ G ¯ of a permutation group G on $$\Omega$$ Ω is defined to be the largest permutation group on $$\Omega$$ Ω , having the same orbits on $$\Omega \times \Omega$$ Ω × Ω as G . It is proved that if G is supersolvable, then $$\overline{G}$$ G ¯ can be found in polynomial time in $$|\Omega|$$ | Ω | . As a by-product of our technique, it is shown that the composition factors
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Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation Comput. Complex. (IF 1.4) Pub Date : 2020-06-01 Or Meir
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $$\textbf{P} \not\subseteq \textbf{NC}^{1}$$ P ⊈ 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 ◊
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Correction to: Communication complexity with small advantage Comput. Complex. (IF 1.4) Pub Date : 2020-05-30 Thomas Watson
Authors would like to correct the incorrect author references in the online published article.
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Communication Complexity with Small Advantage Comput. Complex. (IF 1.4) Pub Date : 2020-04-20 Thomas Watson
We study problems in randomized communication complexity when the protocol is only required to attain some small advantage over purely random guessing, i.e., it produces the correct output with probability at least $$\epsilon$$ ϵ greater than one over the codomain size of the function. Previously, Braverman and Moitra (in: Proceedings of the 45th symposium on theory of computing (STOC), ACM, pp 161–170
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On the characterization of 1-sided error strongly testable graph properties for bounded-degree graphs Comput. Complex. (IF 1.4) Pub Date : 2020-01-20 Hiro Ito, Areej Khoury, Ilan Newman
We study property testing of (di)graph properties in bounded-degree graph models. The study of graph properties in bounded-degree models is one of the focal directions of research in property testing in the last 15 years. However, despite the many results and the extensive research effort, there is no characterization of the properties that are strongly testable (i.e. testable with constant query complexity)
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Simulation Theorems via Pseudo-random Properties Comput. Complex. (IF 1.4) Pub Date : 2019-07-18 Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, Sagnik Mukhopadhyay
We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999), to any gadget which satisfies a certain hitting property. We prove that inner product and gap-Hamming satisfy this property, and as a corollary, we obtain a deterministic simulation theorem for these gadgets, where the gadget’s input size is logarithmic in the input size of the outer function.
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Average-case linear matrix factorization and reconstruction of low width algebraic branching programs Comput. Complex. (IF 1.4) Pub Date : 2019-07-18 Neeraj Kayal, Vineet Nair, Chandan Saha
AbstractA matrix X is called a linear matrix if its entries are affine forms, i.e., degree one polynomials in n variables. What is a minimal-sized representation of a given matrix F as a product of linear matrices? Finding such a minimal representation is closely related to finding an optimal way to compute a given polynomial via an algebraic branching program. Here we devise an efficient algorithm
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A quadratic lower bound for homogeneous algebraic branching programs Comput. Complex. (IF 1.4) Pub Date : 2019-06-08 Mrinal Kumar
AbstractAn algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in $$\mathbb{F}[x_1, x_2, \ldots , x_n]$$F[x1,x2,…,xn]. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path