Randomized Communication and Implicit Representations for Matrices and Graphs of Small Sign-Rank

We prove a characterization of the structural conditions on matrices of sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin randomized communication protocols. Therefore, under these conditions, these graphs also admit implicit representations.

The sign-rank of a matrix M ∈ ±1 N ×N is the smallest rank of a matrix R such that M i,j = sign(R i,j ) for all i, j ∈ [N ]; equivalently, it is the smallest dimension d in which M can be represented as a point-halfspace incidence matrix with halfspaces through the origin, and it is essentially equivalent to the unbounded-error communication complexity. Matrices of sign-rank 3 can achieve the maximum possible bounded-error randomized communication complexity Θ(log N ), and meanwhile the existence of implicit representations for graphs of bounded sign-rank (including UDGs, which have sign-rank 4) has been open since at least 2003. We prove that matrices of signrank 3, and UDGs, have constant randomized communication complexity if and only if they do not encode arbitrarily large instances of the Greater-Than communication problem, or, equivalently, if they do not contain large half-graphs as semi-induced subgraphs. This also establishes the existence of implicit representations for these graphs under the same conditions.