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  • On $$\epsilon$$ ϵ -sensitive monotone computations
    Comput. Complex. (IF 0.85) 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

    更新日期:2020-07-25
  • Two-closures of supersolvable permutation groups in polynomial time
    Comput. Complex. (IF 0.85) Pub Date : 2020-06-24
    Ilia Ponomarenko; Andrey Vasil’ev

    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

    更新日期:2020-06-24
  • Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation
    Comput. Complex. (IF 0.85) Pub Date : 2020-06-06
    Or Meir

    One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., \(\textbf{P} \not\subseteq \textbf{NC}^{1}\)). 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

    更新日期:2020-06-06
  • Correction to: Communication complexity with small advantage
    Comput. Complex. (IF 0.85) Pub Date : 2020-05-30
    Thomas Watson

    Authors would like to correct the incorrect author references in the online published article.

    更新日期:2020-05-30
  • Communication Complexity with Small Advantage
    Comput. Complex. (IF 0.85) 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 \(\epsilon\) 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

    更新日期:2020-04-20
  • On the characterization of 1-sided error strongly testable graph properties for bounded-degree graphs
    Comput. Complex. (IF 0.85) 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)

    更新日期:2020-01-20
  • Simulation Theorems via Pseudo-random Properties
    Comput. Complex. (IF 0.85) 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.

    更新日期:2019-07-18
  • Average-case linear matrix factorization and reconstruction of low width algebraic branching programs
    Comput. Complex. (IF 0.85) Pub Date : 2019-07-18
    Neeraj Kayal; Vineet Nair; Chandan Saha

    A 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 for an

    更新日期:2019-07-18
  • A quadratic lower bound for homogeneous algebraic branching programs
    Comput. Complex. (IF 0.85) Pub Date : 2019-06-08
    Mrinal Kumar

    An 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 \(\mathbb{F}[x_1, x_2, \ldots , x_n]\). 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. An ABP is said to be

    更新日期:2019-06-08
  • On Derandomized Composition of Boolean Functions
    Comput. Complex. (IF 0.85) Pub Date : 2019-06-06
    Or Meir

    The (block-)composition of two Boolean functions \(f : \{0, 1\}^{m} \rightarrow \{0, 1\}, g : \{0, 1\}^{n} \rightarrow \{0, 1\}\) is the function \(f \diamond g\) that takes as inputs m strings \(x_{1}, \ldots , x_{m} \in \{0, 1\}^{n}\) and computes$$(f \diamond g)(x_{1}, \ldots , x_{m}) = f (g(x_{1}), \ldots , g(x_{m})).$$This operation has been used several times in the past for amplifying different

    更新日期:2019-06-06
  • Hierarchy Theorems for Testing Properties in Size-Oblivious Query Complexity
    Comput. Complex. (IF 0.85) Pub Date : 2019-06-06
    Oded Goldreich

    Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function \(q : (0, 1] \rightarrow \mathbb{N}\), we prove the existence of properties for which \(\epsilon\)-testing has query

    更新日期:2019-06-06
  • Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach
    Comput. Complex. (IF 0.85) Pub Date : 2019-05-28
    Suryajith Chillara; Partha Mukhopadhyay

    Tavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any \(n^{O(1)}\)-variate and degree n polynomial in \(\mathsf {VP}\) can be computed by a depth-4 \(\Sigma \Pi \Sigma \Pi \) circuit of size \(2^{O(\sqrt{n}\log n)}\). So, to prove \(\mathsf {VP}\ne \mathsf {VNP}\) it is sufficient to show that an explicit polynomial in \(\mathsf {VNP}\) of degree

    更新日期:2019-05-28
  • Improved Bounds for Quantified Derandomization of Constant-Depth Circuits and Polynomials
    Comput. Complex. (IF 0.85) Pub Date : 2019-04-22
    Roei Tell

    This work studies the question of quantified derandomization, which was introduced by Goldreich and Wigderson (STOC 2014). The generic quantified derandomization problem is the following: For a circuit class \({\mathcal{C}}\) and a parameter B=B(n), given a circuit \({C\in\mathcal{C}}\) with n input bits, decide whether C rejects all of its inputs, or accepts all but B(n) of its inputs. In the current

    更新日期:2019-04-22
  • Random resolution refutations
    Comput. Complex. (IF 0.85) Pub Date : 2019-04-22
    Pavel Pudlák; Neil Thapen

    We study the random resolution refutation system defined in Buss et al. (J Symb Logic 79(2):496–525, 2014). This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if \({{\bf P} \neq {\bf NP}}\), then random resolution cannot

    更新日期:2019-04-22
  • A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights
    Comput. Complex. (IF 0.85) Pub Date : 2019-04-22
    Jin-Yi Cai; Xi Chen

    The complexity of graph homomorphism problems has been the subject of intense study for some years. In this paper, we prove a decidable complexity dichotomy theorem for the partition function of directed graph homomorphisms. Our theorem applies to all non-negative weighted forms of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function ZA(G) of directed graph

    更新日期:2019-04-22
  • Vanishing of Littlewood–Richardson polynomials is in P
    Comput. Complex. (IF 0.85) Pub Date : 2019-04-19
    Anshul Adve; Colleen Robichaux; Alexander Yong

    J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a

    更新日期:2019-04-19
  • Correction to: Query-to-Communication Lifting for P NP
    Comput. Complex. (IF 0.85) Pub Date : 2019-04-06
    Mika Göös, Pritish Kamath, Toniann Pitassi, Thomas Watson

    Unfortunately, the inline image was not processed in the original version and the images are updated here. The original article has been corrected.

    更新日期:2019-04-06
  • Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds
    Comput. Complex. (IF 0.85) Pub Date : 2019-04-06
    Or Meir; Avi Wigderson

    Consider a random sequence of n bits that has entropy at least n−k, where \({k\ll n}\) . A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query \({\approx\frac{n}{k}}\)

    更新日期:2019-04-06
  • Query-to-Communication Lifting for P NP
    Comput. Complex. (IF 0.85) Pub Date : 2018-11-30
    Mika Göös; Pritish Kamath; Toniann Pitassi; Thomas Watson

    We prove that the PNP-type query complexity (alternatively, decision list width) of any Boolean function f is quadratically related to the PNP-type communication complexity of a lifted version of f. As an application, we show that a certain “product” lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture PNP communication complexity up to polynomial factors, which answers a question

    更新日期:2018-11-30
  • Approximate Nonnegative Rank is Equivalent to the Smooth Rectangle Bound
    Comput. Complex. (IF 0.85) Pub Date : 2018-11-20
    Gillat Kol; Shay Moran; Amir Shpilka; Amir Yehudayoff

    We consider two known lower bounds on randomized communication complexity: the smooth rectangle bound and the logarithm of the approximate nonnegative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term.The logarithm of the nonnegative rank is known to be a nearly tight lower bound on the deterministic communication complexity. Our result indicates

    更新日期:2018-11-20
  • Interactive proofs and a Shamir-like result for real number computations
    Comput. Complex. (IF 0.85) Pub Date : 2018-11-07
    Martijn Baartse; Klaus Meer

    We introduce and study interactive proofs in the framework of real number computations as introduced by Blum, Shub, and Smale. Ivanov and de Rougemont started this line of research showing that an analogue of Shamir’s result holds in the real additive Blum–Shub–Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model in which

    更新日期:2018-11-07
  • Lower Bounds and PIT for Non-commutative Arithmetic Circuits with Restricted Parse Trees
    Comput. Complex. (IF 0.85) Pub Date : 2018-09-29
    Guillaume Lagarde; Nutan Limaye; Srikanth Srinivasan

    We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring \({\mathbb{F}\langle{x_1,\ldots,x_N\rangle}}\), where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions

    更新日期:2018-09-29
  • Asymptotic tensor rank of graph tensors: beyond matrix multiplication
    Comput. Complex. (IF 0.85) Pub Date : 2018-09-29
    Matthias Christandl; Péter Vrana; Jeroen Zuiddam

    We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For \({k \geq 4}\), we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the

    更新日期:2018-09-29
  • On the Relationship Between Statistical Zero-Knowledge and Statistical Randomized Encodings
    Comput. Complex. (IF 0.85) Pub Date : 2018-08-20
    Benny Applebaum; Pavel Raykov

    Statistical Zero-knowledge proofs (Goldwasser et al. in SICOMP: SIAM J Comput, 1989) allow a computationally unbounded server to convince a computationally limited client that an input x is in a language \({\Pi}\) without revealing any additional information about x that the client cannot compute by herself. Randomized encoding (RE) of functions (Ishai & Kushilevitz in FOCS 2000) allows a computationally

    更新日期:2018-08-20
  • On semiring complexity of Schur polynomials
    Comput. Complex. (IF 0.85) Pub Date : 2018-06-04
    Sergey Fomin; Dima Grigoriev; Dorian Nogneng; Éric Schost

    Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial \({s_\lambda(x_1,\dots,x_k)}\) labeled by a partition \({\lambda=(\lambda_1\ge\lambda_2\ge\cdots)}\) is bounded by \({O(\log(\lambda_1))}\) provided the number of variables k is fixed.

    更新日期:2018-06-04
  • An adaptivity hierarchy theorem for property testing
    Comput. Complex. (IF 0.85) Pub Date : 2018-05-24
    Clément L. Canonne; Tom Gur

    Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries.In this work, we

    更新日期:2018-05-24
  • Algebraic independence over positive characteristic: New criterion and applications to locally low-algebraic-rank circuits
    Comput. Complex. (IF 0.85) Pub Date : 2018-05-14
    Anurag Pandey; Nitin Saxena; Amit Sinhababu

    The motivation for this work (Pandey et al. 2016) comes from two problems: testing algebraic independence of arithmetic circuits over a field of small characteristic and generalizing the structural property of algebraic dependence used by Kumar, Saraf, CCC’16 to arbitrary fields. It is known that in the case of zero, or large characteristic, using a classical criterion based on the Jacobian, we get

    更新日期:2018-05-14
  • Tensor surgery and tensor rank
    Comput. Complex. (IF 0.85) Pub Date : 2018-03-22
    Matthias Christandl; Jeroen Zuiddam

    We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition

    更新日期:2018-03-22
  • The Landscape of Communication Complexity Classes
    Comput. Complex. (IF 0.85) Pub Date : 2018-03-22
    Mika Göös; Toniann Pitassi; Thomas Watson

    We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between \({\mathsf{P}}\) and \({\mathsf{PSPACE}}\), short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication

    更新日期:2018-03-22
  • Constructive non-commutative rank computation is in deterministic polynomial time
    Comput. Complex. (IF 0.85) Pub Date : 2018-03-22
    Gábor Ivanyos; Youming Qiao; K. V. Subrahmanyam

    We extend the techniques developed in Ivanyos et al. (Comput Complex 26(3):717–763, 2017) to obtain a deterministic polynomial-time algorithm for computing the non-commutative rank of linear spaces of matrices over any field.The key new idea that causes a reduction in the time complexity of the algorithm in Ivanyos et al. (2017) from exponential time to polynomial time is a reduction procedure that

    更新日期:2018-03-22
  • On Space and Depth in Resolution
    Comput. Complex. (IF 0.85) Pub Date : 2017-10-17
    Alexander Razborov

    We show that the total space in resolution, as well as in any other reasonable proof system, is equal (up to a polynomial and \({(\log n)^{O(1)}}\) factors) to the minimum refutation depth. In particular, all these variants of total space are equivalent in this sense. The same conclusion holds for variable space as long as we penalize for excessively (that is, super-exponential) long proofs, which

    更新日期:2017-10-17
  • The Complexity of Approximating complex-valued Ising and Tutte partition functions
    Comput. Complex. (IF 0.85) Pub Date : 2017-09-13
    Leslie Ann Goldberg; Heng Guo

    We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters. Our results are partly motivated by the study of the quantum complexity classes BQP and IQP. Recent results show how to encode quantum computations as evaluations of classical partition functions. These results rely on interesting and deep results about quantum computation in order to

    更新日期:2017-09-13
  • Communication with Contextual Uncertainty
    Comput. Complex. (IF 0.85) Pub Date : 2017-08-29
    Badih Ghazi; Ilan Komargodski; Pravesh K. Kothari; Madhu Sudan

    We introduce a simple model illustrating the utility of context in compressing communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information \({X \in \{0,1\}^n}\) and Bob gets \({Y \in \{0,1\}^n}\), where (X, Y) is drawn from a known distribution, and Bob wishes to compute some function

    更新日期:2017-08-29
  • Some observations on holographic algorithms
    Comput. Complex. (IF 0.85) Pub Date : 2017-08-29
    Leslie G. Valiant

    We define the notion of diversity for families of finite functions and express the limitations of a simple class of holographic algorithms, called elementary algorithms, in terms of limitations on diversity. We show that this class of elementary algorithms is too weak to solve the Boolean circuit value problem, or Boolean satisfiability, or the permanent. The lower bound argument is a natural but apparently

    更新日期:2017-08-29
  • On vanishing of Kronecker coefficients
    Comput. Complex. (IF 0.85) Pub Date : 2017-07-27
    Christian Ikenmeyer; Ketan D. Mulmuley; Michael Walter

    We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood–Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood–Richardson coefficients, unless P = NP.We also show that there exists a #P-formula for a particular subclass

    更新日期:2017-07-27
  • Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity
    Comput. Complex. (IF 0.85) Pub Date : 2017-07-27
    Irit Dinur; Or Meir

    One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de Morgan formulas. Karchmer et al. (Comput Complex 5(3/4):191–204, 1995b) suggested to approach this problem by proving that formula complexity behaves “as expected” with respect to the composition of functions \({f\diamond g}\) . They showed that this conjecture, if proved, would imply super-polynomial

    更新日期:2017-07-27
  • Autoreducibility of NP-Complete Sets under Strong Hypotheses
    Comput. Complex. (IF 0.85) Pub Date : 2017-07-18
    John M. Hitchcock; Hadi Shafei

    We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: ◦ For every \({k \geq 2}\), there is a k-T-complete set for NP that is k-T-autoreducible, but is not k-tt-autoreducible or (k − 1)-T-autoreducible. ◦ For every \({k \geq 3}\), there is a k-tt-complete set for NP that

    更新日期:2017-07-18
  • The Average Sensitivity of Bounded-Depth Formulas
    Comput. Complex. (IF 0.85) Pub Date : 2017-07-05
    Benjamin Rossman

    We show that unbounded fan-in Boolean formulas of depth d + 1 and size s have average sensitivity \({O(\frac{1}{d} \log s)^d}\). In particular, this gives a tight \({2^{\Omega(d(n^{1/d}-1))}}\) lower bound on the size of depth d + 1 formulas computing the parity function. These results strengthen the corresponding \({2^{\Omega(n^{1/d})}}\) and \({O(\log s)^d}\) bounds for circuits due to Håstad (Proceedings

    更新日期:2017-07-05
  • Local Expanders
    Comput. Complex. (IF 0.85) Pub Date : 2017-06-20
    Emanuele Viola; Avi Wigderson

    A map \({f : \{0,1\}^{n} \to \{0,1\}^{n}}\) has locality t if every output bit of f depends only on t input bits. Arora et al. (Colloquium on automata, languages and programming, ICALP, 2009) asked if there exist bounded-degree expander graphs on 2n nodes such that the neighbors of a node \({x\in\{0,1\}^{n}}\) can be computed by maps of constant locality. We give an explicit construction of such graphs

    更新日期:2017-06-20
  • Short lists with short programs in short time
    Comput. Complex. (IF 0.85) Pub Date : 2017-04-19
    Bruno Bauwens; Anton Makhlin; Nikolay Vereshchagin; Marius Zimand

    Given a machine U, a c-short program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any standard Turing machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a \({\operatorname{O}\bigl({\mathrm{log}}|x|\bigr)}\)-short program for x. We also show that there

    更新日期:2017-04-19
  • Tight Size-Degree Bounds for Sums-of-Squares Proofs
    Comput. Complex. (IF 0.85) Pub Date : 2017-04-19
    Massimo Lauria; Jakob Nordström

    We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size \({n^{\Omega{(d)}}}\) for values of d = d(n) from constant all the way up to \({n^{\delta}}\) for some universal constant \({\delta}\). This shows that the \({{n^{{\rm O}{(d)}}}}\) running time obtained by using the Lasserre semidefinite

    更新日期:2017-04-19
  • Block-symmetric polynomials correlate with parity better than symmetric
    Comput. Complex. (IF 0.85) Pub Date : 2017-04-05
    Frederic Green; Daniel Kreymer; Emanuele Viola

    We show that degree-d block-symmetric polynomials in n variables modulo any odd p correlate with parity exponentially better than degree-d symmetric polynomials, if \({n \ge cd^2 {\rm log} d}\) and \({d \in [0.995 \cdot p^t - 1,p^t)}\) for some \({t \ge 1}\) and some \({c > 0}\) that depends only on p. For these infinitely many degrees, our result solves an open problem raised by a number of researchers

    更新日期:2017-04-05
  • List-Decoding Barnes–Wall Lattices
    Comput. Complex. (IF 0.85) Pub Date : 2017-01-12
    Elena Grigorescu; Chris Peikert

    The question of list-decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in \({\mathbb{R}^{N}}\), and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely:

    更新日期:2017-01-12
  • Graph Isomorphism, Color Refinement, and Compactness
    Comput. Complex. (IF 0.85) Pub Date : 2016-12-19
    V. Arvind; Johannes Köbler; Gaurav Rattan; Oleg Verbitsky

    Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai et al. (SIAM J Comput 9(3):628–635, 1980) have shown that random graphs are amenable

    更新日期:2016-12-19
  • Low-degree test with polynomially small error
    Comput. Complex. (IF 0.85) Pub Date : 2016-12-19
    Dana Moshkovitz

    A long line of work in Theoretical Computer Science shows that a function is close to a low-degree polynomial iff it is locally close to a low-degree polynomial. This is known as low-degree testing and is the core of the algebraic approach to construction of PCP. We obtain a low-degree test whose error, i.e., the probability it accepts a function that does not correspond to a low-degree polynomial

    更新日期:2016-12-19
  • On Approximating the Eigenvalues of Stochastic Matrices in Probabilistic Logspace
    Comput. Complex. (IF 0.85) Pub Date : 2016-12-16
    Dean Doron; Amir Sarid; Amnon Ta-Shma

    We show that approximating the second eigenvalue of stochastic operators is BPL-complete, thus giving a natural problem complete for this class. We also show that approximating any eigenvalue of a stochastic and Hermitian operator with constant accuracy can be done in BPL. This work together with related work on the subject reveal a picture where the various space-bounded classes (e.g., probabilistic

    更新日期:2016-12-16
  • On the hardness of the noncommutative determinant
    Comput. Complex. (IF 0.85) Pub Date : 2016-11-30
    V. Arvind; Srikanth Srinivasan

    In this paper, we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: ○ We show that if the noncommutative

    更新日期:2016-11-30
  • Dual VP Classes
    Comput. Complex. (IF 0.85) Pub Date : 2016-09-23
    Eric Allender; Anna Gál; Ian Mertz

    We consider the complexity class ACC 1 and related families of arithmetic circuits. We prove a variety of collapse results, showing several settings in which no loss of computational power results if fan-in of gates is severely restricted, as well as presenting a natural class of arithmetic circuits in which no expressive power is lost by severely restricting the algebraic degree of the circuits. We

    更新日期:2016-09-23
  • Matrix rigidity of random Toeplitz matrices
    Comput. Complex. (IF 0.85) Pub Date : 2016-09-01
    Oded Goldreich; Avishay Tal

    A matrix A is said to have rigidity s for rank r if A differs from any matrix of rank r on more than s entries. We prove that random n-by-n Toeplitz matrices over \({\mathbb{F}_{2}}\) (i.e., matrices of the form \({A_{i,j} = a_{i-j}}\) for random bits \({a_{-(n-1)}, \ldots, a_{n-1}}\)) have rigidity \({\Omega(n^3/(r^2\log n))}\) for rank \({r \ge \sqrt{n}}\), with high probability. This improves, for

    更新日期:2016-09-01
  • Topological lower bounds for arithmetic networks
    Comput. Complex. (IF 0.85) Pub Date : 2016-09-01
    Andrei Gabrielov; Nicolai Vorobjov

    We prove that the depth of any arithmetic network for deciding membership in a semialgebraic set \({\Sigma \subset \mathbb{R}^{n}}\) is bounded from below by$$c_1 \sqrt{ \frac{\log ({\rm b}(\Sigma))}{n}} -c_2 \log n,$$where \({{\rm b}(\Sigma)}\) is the sum of the Betti numbers of \({\Sigma}\) with respect to “ordinary” (singular) homology, and c 1, c 2 are some (absolute) positive constants. This result

    更新日期:2016-09-01
  • Non-commutative Edmonds’ problem and matrix semi-invariants
    Comput. Complex. (IF 0.85) Pub Date : 2016-08-19
    Gábor Ivanyos; Youming Qiao; K. V. Subrahmanyam

    In 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an \({n \times n}\) matrix T with integral homogeneous linear polynomials. In this paper, we consider the non-commutative version of Edmonds’ problem: compute the rank of T over the free skew field. This problem has been proposed, sometimes in disguise, from several different perspectives in the study

    更新日期:2016-08-19
  • Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs
    Comput. Complex. (IF 0.85) Pub Date : 2016-08-02
    Rohit Gurjar; Arpita Korwar; Nitin Saxena; Thomas Thierauf

    A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial-time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial-time complexity \({n^{O({\rm log}\,n)}}\). In both the cases, our

    更新日期:2016-08-02
  • Sparse multivariate polynomial interpolation on the basis of Schubert polynomials
    Comput. Complex. (IF 0.85) Pub Date : 2016-07-12
    Priyanka Mukhopadhyay; Youming Qiao

    Schubert polynomials were discovered by A. Lascoux and M. Schützenberger in the study of cohomology rings of flag manifolds in 1980s. These polynomials generalize Schur polynomials and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials and expand in the Schubert basis with the generalized Littlewood–Richardson

    更新日期:2016-07-12
  • Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits
    Comput. Complex. (IF 0.85) Pub Date : 2016-06-23
    Yasuhiro Takahashi; Seiichiro Tani

    We study the quantum complexity class \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\) of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\), which is an affirmative answer to the question posed by Høyer and Špalek. In sharp contrast to the strict

    更新日期:2016-06-23
  • Quantum Query Complexity of Almost All Functions with Fixed On-set Size
    Comput. Complex. (IF 0.85) Pub Date : 2016-06-21
    Andris Ambainis; Kazuo Iwama; Masaki Nakanishi; Harumichi Nishimura; Rudy Raymond; Seiichiro Tani; Shigeru Yamashita

    This paper considers the quantum query complexity of almost all functions in the set \({\mathcal{F}}_{N,M}\) of \({N}\)-variable Boolean functions with on-set size \({M (1\le M \le 2^{N}/2)}\), where the on-set size is the number of inputs on which the function is true. The main result is that, for all functions in \({\mathcal{F}}_{N,M}\) except its polynomially small fraction, the quantum query complexity

    更新日期:2016-06-21
  • The Value of Help Bits in Randomized and Average-Case Complexity
    Comput. Complex. (IF 0.85) Pub Date : 2016-06-09
    Salman Beigi; Omid Etesami; Amin Gohari

    “Help bits" are some limited trusted information about an instance or instances of a computational problem that may reduce the computational complexity of solving that instance or instances. In this paper, we study the value of help bits in the settings of randomized and average-case complexity.If k instances of a decision problem can be efficiently solved using \({\ell < k}\) help bits, then without

    更新日期:2016-06-09
  • Non-interactive proofs of proximity
    Comput. Complex. (IF 0.85) Pub Date : 2016-06-03
    Tom Gur; Ron D. Rothblum

    We initiate a study of non-interactive proofs of proximity. These proof systems consist of a verifier that wishes to ascertain the validity of a given statement, using a short (sublinear length) explicitly given proof, and a sublinear number of queries to its input. Since the verifier cannot even read the entire input, we only require it to reject inputs that are far from being valid. Thus, the verifier

    更新日期:2016-06-03
  • On the connection between interval size functions and path counting
    Comput. Complex. (IF 0.85) Pub Date : 2016-06-02
    Evangelos Bampas; Andreas-Nikolas Göbel; Aris Pagourtzis; Aris Tentes

    We investigate the complexity of hard (#P-complete) counting problems that have easy decision version. By ‘easy decision,’ we mean that deciding whether the result of counting is nonzero is in P. This property is shared by several well-known problems, such as counting the number of perfect matchings in a given graph or counting the number of satisfying assignments of a given DNF formula. We focus on

    更新日期:2016-06-02
  • Sunflowers and Testing Triangle-Freeness of Functions
    Comput. Complex. (IF 0.85) Pub Date : 2016-05-30
    Ishay Haviv; Ning Xie

    A function \({f : {\mathbb F}_{2}^{n} \rightarrow {\{0,1\}}}\) is triangle-free if there are no \({x_{1},x_{2},x_{3} \in {\mathbb F}_{2}^{n}}\) satisfying \({x_{1} + x_{2} + x_{3} = 0}\) and \({f(x_{1}) = f(x_{2}) = f(x_{3}) = 1}\). In testing triangle-freeness, the goal is to distinguish with high probability triangle-free functions from those that are \({\varepsilon}\)-far from being triangle-free

    更新日期:2016-05-30
  • Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin
    Comput. Complex. (IF 0.85) Pub Date : 2016-05-30
    Neeraj Kayal; Chandan Saha

    Shpilka & Wigderson (IEEE conference on computational complexity, vol 87, 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth-three arithmetic circuits with bounded bottom fanin over a field \({{\mathbb{F}}}\) of characteristic zero. We resolve this problem by proving a \({N^{\Omega(\frac{d}{\tau})}}\) lower bound for (nonhomogeneous) depth-three arithmetic circuits

    更新日期:2016-05-30
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