# Towards a complexity theory for the congested clique

## TL;DR

This paper develops a complexity theory framework for the congested clique model, introducing new classes, hierarchy theorems, and relationships between problems to better understand its computational limits.

## Contribution

It introduces nondeterministic and hierarchy models, proves lower bounds, and maps problem relationships, advancing the theoretical understanding of congested clique complexity.

## Key findings

- Defined the nondeterministic congested clique model.
- Proved a time hierarchy theorem for the model.
- Mapped relationships between natural problems in the model.

## Abstract

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now know that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity-theoretic tools to build a clearer picture of the complexity landscape of the congested clique: -- Nondeterminism and beyond: We introduce the nondeterministic congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms. We further generalise these notions by introducing the constant-round decision hierarchy (analogous to the polynomial hierarchy). -- Non-constructive lower bounds: We lift the prior non-uniform counting arguments to a general technique for proving non-constructive uniform lower bounds for the congested clique. In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings. -- Fine-grained complexity: We map out relationships between various natural problems in the congested clique model, arguing that a reduction-based complexity theory currently gives us a fairly good picture of the complexity landscape of the congested clique.