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Disjointness is Hard in the Multiparty Number-on-the-Forehead Model

Troy Lee, Adi Shraibman

2009
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Computational Complexity
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We show that disjointness requires randomized communication Ω n 1/(k+1) 2 2 k in the general k-party number-on-the-forehead model of complexity. The previous best lower bound for k ≥ 3 was log n k−1 . Our results give a separation between nondeterministic and randomized multiparty number-on-the-forehead communication complexity for up to k = log log n−O(log log log n) many players. Also by a reduction of Beame, Pitassi, and Segerlind, these results imply subexponential lower bounds on the size
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... f proofs needed to refute certain unsatisfiable CNFs in a broad class of proof systems, including tree-like Lovász-Schrijver proofs. consequences of such lower bounds: for example, by results of [HG91, BT94] , showing a superpolylogarithmic lower bound on an explicit function for polylogarithmic many players would give an explicit function outside of the class ACC 0 -that is, a function which requires superpolynomial size constant-depth circuits using AND, OR, NOT, and modulo m gates. While showing such bounds remains a challenging open problem, we do know of explicit functions which require large communication in this model for Θ(log n) many players. Babai, Nisan, and Szegedy [BNS89] showed that the inner product function generalized to k-parties requires randomized communication Ω(n/4 k ), and for other explicit functions slightly larger bounds of size Ω(n/2 k ) are known [FG05] . These lower bounds are all achieved using the discrepancy method, a very general technique which gives lower bounds even on randomized models with error probability close to 1/2, and also on nondeterministic communication complexity. For some basic functions, however, there is a huge gap in our knowledge. One example is the disjointness function, or equivalently its complement, set intersection. In the set intersection problem, the goal of the players is to determine if there is an index j such that every string x i has a −1 in position j, where here and throughout the paper we interpret −1 as 'true.' The best known protocol has cost O(k 2 n log(n)/2 k ) [Gro94]. On the other hand, the best lower bound in the general number-on-the-forehead model is log n k−1 , for k ≥ 3 [Tes02, BPSW06]. For k = 2 tight bounds are known of Θ(n) for randomized communication complexity [KS87] and Θ( √ n) for quantum communication complexity [Raz03, AA05]. A major obstacle toward proving better lower bounds on set intersection is that it has a low cost nondeterministic protocol. In case there is a position where all players have a −1, with O(log n) bits a prover can send the name of this position and the players can then verify this is the case. Since the discrepancy method is also a lower bound on nondeterministic complexity, it is limited to logarithmic lower bounds for set intersection. Even in the two-party case, determining the complexity of set intersection in the randomized and quantum models was a long-standing open problem, in part for this reason. In the multiparty case, the discrepancy method is the only technique which has been used to show lower bounds on the general randomized model of number-on-the-forehead complexity. Although other two-party methods can be generalized to the multiparty number-on-the-forehead model, they can become very difficult to handle. One source of this difficulty is that, whereas in the two party case we can nicely represent the function f (x, y) as a matrix, in the multiparty case we deal with higher dimensional tensors. This makes many of the linear algebraic tools so useful in the two-party case inapplicable or at least much more involved. For example, while matrix rank is a staple lower bound technique for deterministic two-party complexity, in the tensor case even basic questions like the maximum rank of a n × n × n tensor remain open. Besides this technical challenge, additional motivation to studying the number-on-the-forehead complexity of disjointness was given by Beame, Pitassi, and Segerlind [BPS06], who showed that lower bounds on disjointness imply lower bounds on a very general class of proof systems, including cutting planes and Lovász-Schrijver proof systems. We show that disjointness requires randomized communication Ω n 1/(k+1) 2 2 k in the general kparty number-on-the-forehead model. This separates nondeterministic and randomized multiparty number-on-the-forehead complexity for up to k = log log n − O(log log log n) many players. Also

doi:10.1007/s00037-009-0276-2
fatcat:ovvpwvx6s5g2fncwviztxd5ice