# Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model

## TL;DR

This paper establishes the first super-linear lower bounds for fundamental graph problems in the CONGEST model, revealing inherent communication complexity limitations for exact solutions in distributed computing.

## Contribution

It provides the first super-linear lower bounds for natural graph problems in the CONGEST model, including NP-hard and polynomial problems, and explores bounds for weighted all-pairs-shortest-paths.

## Key findings

- Minimum vertex cover requires (n^2/log^2 n) rounds
- Graph -coloring requires (n^2/log^2 n) rounds
- Weighted all-pairs-shortest-paths has an (n) lower bound

## Abstract

We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires $\Omega(n^2/\log^2{n})$ rounds in the worst case in the CONGEST model, as well as any algorithm for $\chi$-coloring a graph, where $\chi$ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple $\Omega(n)$ lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a complexity of $\Theta(n/\log{n})$. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.