Results 1  10
of
18
Property Testing Lower Bounds Via Communication Complexity
, 2011
"... We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexit ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is klinear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with twosided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as kjuntas. For some classes, such as the class of monotone functions and the class of ssparse GF(2) polynomials, we significantly strengthen the best known bounds.
Systems of linear equations over F2 and problems parameterized above average
 Proc. SWAT 2010
, 1999
"... In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over F2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over F2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC’06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least k, where k is the parameter. It is not hard to see that we may assume that no two equations in Az = b have the same lefthand side and n = rankA. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixedparameter
Quantum Boolean Functions
, 2009
"... In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the GoldreichLevin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of
On the communication complexity of XOR functions
, 2010
"... An XOR function is a function of the form g(x, y) = f(x ⊕ y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise oneway communication complexity for all f. We also show that ..."
Abstract
 Add to MetaCart
An XOR function is a function of the form g(x, y) = f(x ⊕ y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise oneway communication complexity for all f. We also show that, when f is monotone, g’s quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g’s quantum complexity is Θ(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness. 1
The Streaming Complexity of Cycle Counting, Sorting By Reversals, and Other Problems
, 2010
"... In this paper we introduce a new technique for proving streaming lower bounds (and oneway communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a problem that we introduce and prove the first lower bound for, but it is a generalizati ..."
Abstract
 Add to MetaCart
In this paper we introduce a new technique for proving streaming lower bounds (and oneway communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a problem that we introduce and prove the first lower bound for, but it is a generalization of a wellknown problem called the Boolean Hidden Matching, that was used by Gavinsky et al. to prove separations between quantum communication complexity and oneway randomized communication complexity. The hardness of the BHH problem is inherently oneway: it is easy to solve using logarithmic twoway communication, but requires √ n communication if Alice is only allowed to send messages to Bob, and not viceversa. This onewayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently oneway. By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, for approximately counting the number of cycles in a 2regular graph, and for other problems. For example, here is one lower bound that we prove, for a cyclecounting problem: Alice gets a perfect matching EA on a set of n nodes, and Bob gets a perfect matching EB on the same set of nodes. The union EA ∪ EB is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o ( √ n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is “natively ” oneway: With 4 rounds of communication, it is easy to distinguish these two cases. 1
www.theoryofcomputing.org A Separation of NP and coNP in Multiparty Communication Complexity
, 2010
"... Abstract: We prove that coNP � MA in the numberonforehead model of multiparty communication complexity for up to k =(1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k →{0,1} with conondeterministic complexity O(logn) and MerlinArthur comp ..."
Abstract
 Add to MetaCart
Abstract: We prove that coNP � MA in the numberonforehead model of multiparty communication complexity for up to k =(1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k →{0,1} with conondeterministic complexity O(logn) and MerlinArthur complexity nΩ(1). The problem was open for k � 3. As a corollary, we obtain an explicit separation of NP and coNP for up to k =(1−ε)logn players, complementing an independent result by Beame et al. (2010) who separate these classes nonconstructively for up to k = 2 (1−ε)n players. ACM Classification: F.1.3, F.2.3 AMS Classification: 68Q17, 68Q15 Key words and phrases: multiparty communication complexity, nondeterminism, MerlinArthur computations, separations and lower bounds 1
Acknowledgments
"... The investigations were performed at the Centrum Wiskunde & Informatica (CWI) and were supported by Vici grant 639.023.302 from the Netherlands Organization for Scientific Research (NWO). ..."
Abstract
 Add to MetaCart
The investigations were performed at the Centrum Wiskunde & Informatica (CWI) and were supported by Vici grant 639.023.302 from the Netherlands Organization for Scientific Research (NWO).