Hadamard tensors and lower bounds on multiparty communication complexity

Jeff Ford, Anna Gál
2012 Computational Complexity  
We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the "Number on the Forehead" model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any k-party communication problem represented by a Hadamard tensor must have Ω(n/2 k ) multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving Ω(n/2 k ) lower bounds on
more » ... multiparty communication complexity for a new class of explicitly defined Boolean functions. The largest known lower bounds for explicit functions are of the form Ω(n/2 k ) where k is the number of players, and n is the number of bits each player misses. The first bounds of this form were given by Babai, Nisan and Szegedy [2] for the "quadratic character of the sum of coordinates" (QCS) function. They also gave an Ω(n/4 k ) lower bound for the "generalized inner product" (GIP) function that was later improved to Ω(n/2 k ) by Chung and Tetali [10] . Chung [9] and Raz [19] generalized the method of [2] to give a sufficient condition for a function to have Ω(n/2 k ) multiparty communication complexity. Raz [19] also obtained Ω( √ n/2 k ) lower bounds for a new function based upon matrix multiplication over GF(2). Babai, Hayes and Kimmel [3] obtained further examples of functions with Ω(n/2 k ) multiparty communication complexity. All of these lower bounds were obtained by estimating discrepancy, and so they also hold in the distributional and randomized communication complexity models. The known bounds all decrease exponentially as the number of players grows, becoming trivial for k > log n. It is a major open problem, with important implications in circuit complexity, to prove nontrivial lower bounds on multiparty communication problems for a large number of players. The class ACC 0 , defined by Barrington [4], consists of languages recognized by constant depth, unbounded fan-in polynomial size circuit families with AND, OR, NOT and MOD m gates for a fixed m. By the results of [24, 5, 13] , families of functions that belong to ACC 0 can be computed by multiparty protocols with polylogarithmic (in n) communication by a polylogarithmic (in n) number of players (where n is the number of bits each player misses). Separating ACC 0 from other complexity classes (e.g. NP) is a major open problem, and a sufficiently large multiparty communication complexity lower bound would resolve it. As proved by Chor and Goldreich [8], any Boolean function defined by a Hadamard matrix has Ω(n) 2-party communication complexity. Their proof uses a lemma by Lindsey (see [11] p. 88) that estimates the largest possible sum of entries in a submatrix of a Hadamard matrix. Lindsey's lemma implies upper bounds on the discrepancy of functions defined by Hadamard matrices and "nearly" Hadamard matrices. Babai, Nisan and Szegedy [2] generalized the proof of Lindsey's lemma to obtain upper bounds on the discrepancy of tensors associated with certain multiparty communication problems. The lower bounds that followed (e.g. [9, 10, 19, 3] ) all used this approach. These papers did not consider generalizing the Hadamard property to tensors. In fact, [10] mentions that it is not clear how to generalize Hadamard matrices to tensors. In this paper we propose a generalization of the Hadamard property of matrices to tensors of arbitrary dimension. We show that any k-party communication problem represented by a Hadamard tensor must have Ω(n/2 k ) multiparty communication complexity. We construct families of Hadamard tensors, giving Ω(n/2 k ) lower bounds for a new class of explicitly defined Boolean functions. Our Hadamard property is stronger than the sufficient condition of Chung [9] and Raz [19] for Ω(n/2 k ) bounds, and could yield larger than Ω(n/2 k ) lower bounds. There are no matching upper bounds known for functions represented by Hadamard tensors. We show how the Chung-Raz condition and some pre-
doi:10.1007/s00037-012-0052-6 fatcat:er2wewhfmvgujlngi62vsav4ka