• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) Pub Date : 2019-07-18

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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) Pub Date : 2019-05-28

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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
• Comput. Complex. (IF 0.822) 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
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