Satyanarayana V. Lokam

Relations Between Communication Complexity, Linear Arrangements, and Computational Complexity

Recently, Forster [7] proved a new lower bound on probabilistic communication complexity in terms of the operator norm of the communication matrix. In this paper, we want to exploit the various relations between communication complexity of distributed Boolean functions, geometric questions related to half space representations of these functions, and the computational complexity of these functions in various restricted models of computation. In order to widen the range of applicability of Forsters bound, we start with the derivation of a generalized lower bound. We present a concrete family of distributed Boolean functions where the generalized bound leads to a linear lower bound on the probabilistic communication complexity (and thus to an exponential lower bound on the number of Euclidean dimensions needed for a successful half space representation), whereas the old bound fails. We move on to a geometric characterization of the well known communication complexity class C-PP in terms of half space representations achieving a large margin. Our characterization hints to a close connection between the bounded error model of probabilistic communication complexity and the area of large margin classification. In the final section of the paper, we describe how our techniques can be used to prove exponential lower bounds on the size of depth-2 threshold circuits (with still some technical restrictions). Similar results can be obtained for read-k-times randomized ordered binary decision diagram and related models.

Complexity Lower Bounds using Linear Algebra

We survey several techniques for proving lower bounds in Boolean, algebraic, and communication complexity based on certain linear algebraic approaches. The common theme among these approaches is to study robustness measures of matrix rank that capture the complexity in a given model. Suitably strong lower bounds on such robustness functions of explicit matrices lead to important consequences in the corresponding circuit or communication models. Many of the linear algebraic problems arising from these approaches are independently interesting mathematical challenges.

Communication Complexity of Simultaneous Messages

In the multiparty communication game (CFL game) of Chandra, Furst, and Lipton [Proceedings of the 15th Annual ACM Symposium on Theory of Computing, Boston, MA, 1983, pp. 94--99] k players collaboratively evaluate a function f(x0, . . . , xk-1) in which player i knows all inputs except xi. The players have unlimited computational power. The objective is to minimize communication. In this paper, we study the SIMULTANEOUS MESSAGES (SM) model of multiparty communication complexity. The SM model is a restricted version of the CFL game in which the players are not allowed to communicate with each other. Instead, each of the k players simultaneously sends a message to a referee, who sees none of the inputs. The referee then announces the function value. We prove lower and upper bounds on the SM complexity of several classes of explicit functions. Our lower bounds extend to randomized SM complexity via an entropy argument. A lemma establishing a tradeoff between average Hamming distance and range size for transformations of the Boolean cube might be of independent interest. Our lower bounds on SM complexity imply an exponential gap between the SM model and the CFL model for up to

Better lower bounds for locally decodable codes

(\log n)^{1-\epsilon}

Improved Bounds on Security Reductions for Discrete Log Based Signatures

players for any

Width-Based Algorithms for SAT and CIRCUIT-SAT

\epsilon > 0

Graph Complexity and Slice Functions

. This separation is obtained by comparing the respective complexities of the Generalized Addressing Function, GAFG,k, where G is a group of order n. We also combine our lower bounds on SM complexity with the ideas of Hastad and Goldmann [Comput. Complexity, 1 (1991), pp. 113--129] to derive superpolynomial lower bounds for certain depth-2 circuits computing a function related to the GAF function. We prove some counterintuitive upper bounds on SM complexity. We show that {\sf GAF}

Reconstruction of depth-4 multilinear circuits with top fan-in 2

_{\mathbb{Z}_2^t,3}

An optimal lower bound for 2-query locally decodable linear codes

has SM complexity

Efficient Reconstruction of Random Multilinear Formulas

O(n^{0.92})