Inner Product and Set Disjointness: Beyond Logarithmically Many Parties

TL;DR

This paper precisely characterizes the randomized communication complexity of inner product and set disjointness problems in multi-party settings with many parties, revealing thresholds for constant-cost protocols based on the number of parties.

Contribution

It provides exact complexity bounds for these problems for all k ≥ log n, extending understanding beyond logarithmic-party regimes.

Findings

Communication complexity is Θ(1 + (log n)/log(1 + k/log n)) bits.

Constant-cost protocols exist if and only if k ≥ n^ε for some ε > 0.

The results unify and extend previous bounds for multi-party communication complexity.

Abstract

A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f\colon(\{0,1\}^n)^{k}\to\{0,1\}$ with $k\gg\log n$ parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every $k\geq\log n$, showing in both cases that $\Theta(1+\lceil\log n\rceil/\log\lceil1+k/\log n\rceil)$ bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is $k\geq n^{\epsilon}$ for some constant $\epsilon>0.$