Near-optimal edge partitioning via intersecting families

Abstract

We study the problem of edge partitioning, where the goal is to partition the edge set of a graph into $k$ parts. The replication factor of a vertex $v$ is the number of parts that contain edges incident to $v$. The goal is to minimize the average replication factor of the vertices while keeping the sizes of the parts nearly equal. We study the regime where the number of parts is significantly smaller than the size of the graph. To this end, we prove asymptotically tight bounds on the optimal replication factor, both for any constant number of parts $k$ and when $k$ grows slowly with the number of vertices. In particular, for growing $k$, every graph admits an almost balanced partition with average replication factor $\sqrt{k}(1+o(1))$, and this bound is tight. The upper bounds are achieved by a new class of edge partitioning algorithms. These algorithms are computationally efficient, including in the LOCAL and CONGEST models, and can be implemented as stateless streaming algorithms in graph processing frameworks. The lower bounds are witnessed by complete graphs and by jumbled graphs, also known as pseudo-random graphs. Our method generalizes a family of algorithms based on symmetric intersecting families of sets. Informally, we replace the symmetry condition by a weaker balance condition that is still sufficient for the algorithms. This relaxation makes it possible to construct such families with asymptotically optimal rank $\sqrt{k}(1+o(1))$.