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Register a new userDisjointness is hard in the multi-party number on the forehead model
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Added by
Troy Lee
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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We show that disjointness requires randomized communication Omega(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 of proofs needed to refute certain unsatisfiable CNFs in a broad class of proof systems, including tree-like Lovasz-Schrijver proofs.
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Information-theoretic methods have proven to be a very powerful tool in communication complexity, in particular giving an elegant proof of the linear lower bound for the two-party disjointness function, and tight lower bounds on disjointness in the multi-party number-in-the-hand (NIH) model. In this paper, we study the applicability of information theoretic methods to the multi-party number-on-the-forehead model (NOF), where determining the complexity of disjointness remains an important open problem. There are two basic parts to the NIH disjointness lower bound: a direct sum theorem and a lower bound on the one-bit AND function using a beautiful connection between Hellinger distance and protocols revealed by Bar-Yossef, Jayram, Kumar and Sivakumar [BYJKS04]. Inspired by this connection, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party NOF protocols and provide a small toolbox that allows one to manipulate several Hellinger volume terms and lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. We then apply these new tools to obtain a lower bound on the informational complexity of the AND_k function in the NOF setting. Finally, we discuss the difficulties of proving a direct sum theorem for information cost in the NOF model.
We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels. While this model has been widely used in distributed computing, strong lower bounds on the amount of communication needed to compute simple functions have just begun to appear. In this work, we prove a tight lower bound of $Omega(kn)$ on the communication needed for computing the Tribes function, when the underlying graph is a star of $k+1$ nodes that has $k$ leaves with inputs and a center with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds are obtained by building upon the recent information theoretic techniques of Braverman et.al (FOCS13) and combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC03). This approach yields information complexity bounds that is of independent interest.
We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemer{e}di graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemer{e}di. The graphs induced by this protocol have $n$ vertices, $Omega(n^2/log n)$ edges, and are decomposable into $n^{1+O(1/log log n)}$ induced matchings. Another protocol is an explicit (and slightly simpler) version of the construction of Alon, Moitra and Sudakov, producing graphs with similar properties. We also generalize the above protocols to more than three players, in order to construct dense uniform hypergraphs in which every edge lies in a positive small number of simplices.
The computational complexity of a problem arising in the context of sparse optimization is considered, namely, the projection onto the set of $k$-cosparse vectors w.r.t. some given matrix $Omeg$. It is shown that this projection problem is (strongly) NP-hard, even in the special cases in which the matrix $Omeg$ contains only ternary or bipolar coefficients. Interestingly, this is in contrast to the projection onto the set of $k$-sparse vectors, which is trivially solved by keeping only the $k$ largest coefficients.
We prove computational intractability of variants of checkers: (1) deciding whether there is a move that forces the other player to win in one move is NP-complete; (2) checkers where players must always be able to jump on their turn is PSPACE-complete; and (3) cooperati
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