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33
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 ..."
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Cited by 34 (8 self)
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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.
Some applications of hypercontractive inequalities in quantum information
, 2012
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A Separation of NP and coNP in Multiparty Communication Complexity
 THEORY OF COMPUTING
, 2010
"... We prove that coNP � MA and in particular NP ̸ = coNP 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(l ..."
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Cited by 13 (3 self)
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We prove that coNP � MA and in particular NP ̸ = coNP 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.
NearOptimal and Explicit Bell Inequality Violations
"... Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new twoplayer games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Match ..."
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Cited by 12 (3 self)
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Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new twoplayer games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by BarYossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPRpairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log n/ √ n). The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here ndimensional entanglement allows to win the game with probability 1/(logn) 2, while the best winning probability without entanglement is 1/n. This nearlinear ratio (“Bell inequality violation”) is nearoptimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.
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 ..."
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Cited by 10 (4 self)
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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
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 ..."
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Cited by 10 (8 self)
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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
Variations on the Sensitivity Conjecture
 THEORY OF COMPUTING LIBRARY GRADUATE SURVEYS 4 (2011), PP. 1–27
, 2011
"... The sensitivity of a Boolean function f of n Boolean variables is the maximum over all inputs x of the number of positions i such that flipping the ith bit of x changes the value of f (x). Permitting to flip disjoint blocks of bits leads to the notion of block sensitivity, known to be polynomially ..."
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Cited by 6 (0 self)
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The sensitivity of a Boolean function f of n Boolean variables is the maximum over all inputs x of the number of positions i such that flipping the ith bit of x changes the value of f (x). Permitting to flip disjoint blocks of bits leads to the notion of block sensitivity, known to be polynomially related to a number of other complexity measures of f, including the decisiontree complexity, the polynomial degree, and the certificate complexity. A longstanding open question is whether sensitivity also belongs to this equivalence class. A positive answer to this question is known as the Sensitivity Conjecture. We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open. Among other things, we relate the problem to Communication Complexity via recent results by Sherstov (QIC 2010). We also indicate new connections to Fourier analysis.
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 ..."
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Cited by 5 (1 self)
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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
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 ..."
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Cited by 5 (0 self)
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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