On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model

TL;DR

This paper establishes tight lower bounds on the communication complexity of linear algebra problems over finite fields in a multi-player message passing model, using reductions to two-player problems and symmetry properties.

Contribution

It introduces a general framework for deriving multi-player lower bounds from two-player complexities and resolves an open problem on the matrix rank problem's two-player lower bound.

Findings

Proves s-player complexity is at least s times two-player complexity.

Provides new two-player lower bounds for matrix rank and related problems.

Applies bounds to threshold promise versions of problems, relevant for data streaming hardness.

Abstract

We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. Specifically, for a matrix which is distributed among a number of players, we consider the problem of determining its rank, of computing entries in its inverse, and of solving linear equations. We also consider related problems such as computing the generalized inner product of vectors held on different servers. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized $s$-player communication complexity of these problems is at least $s$ times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, which we formally define, we can show the hardest input distribution is a symmetric distribution, and therefore apply a recent multi-player lower bound technique of Phillips et al. Further, we give new two-player lower bounds for a number of these problems. In particular, our optimal lower bound for the two-player version of the matrix rank problem resolves an open question of Sun and Wang. A common feature of our lower bounds is that they apply even to the special "threshold promise" versions of these problems, wherein the underlying quantity, e.g., rank, is promised to be one of just two values, one on each side of some critical threshold. These kinds of promise problems are commonplace in the literature on data streaming as sources of hardness for reductions giving space lower bounds.