Robust Lower Bounds for Graph Problems in the Blackboard Model of Communication

TL;DR

This paper establishes that solving key graph problems in the multi-party blackboard communication model requires linear communication complexity, highlighting fundamental limits in distributed graph algorithms.

Contribution

It provides nearly tight lower bounds of a(n) bits for a wide range of important graph problems in the blackboard communication model.

Findings

Any non-trivial graph problem requires a(n) bits of communication.

Lower bounds are nearly optimal for MIS, Maximal Matching, and a+1-coloring.

Results hold even with random edge distribution among parties.

Abstract

We give lower bounds on the communication complexity of graph problems in the multi-party blackboard model. In this model, the edges of an $n$-vertex input graph are partitioned among $k$ parties, who communicate solely by writing messages on a shared blackboard that is visible to every party. We show that any non-trivial graph problem on $n$-vertex graphs has blackboard communication complexity $\Omega(n)$ bits, even if the edges of the input graph are randomly assigned to the $k$ parties. We say that a graph problem is non-trivial if the output cannot be computed in a model where every party holds at most one edge and no communication is allowed. Our lower bound thus holds for essentially all key graph problems relevant to distributed computing, including Maximal Independent Set (MIS), Maximal Matching, ($\Delta+1$)-coloring, and Dominating Set. In many cases, e.g., MIS, Maximal Matching, and $(\Delta+1)$-coloring, our lower bounds are optimal, up to poly-logarithmic factors.