Tight Bounds for the Min-Max Boundary Decomposition Cost of Weighted Graphs

TL;DR

This paper establishes tight bounds for partitioning weighted graphs with bounded degree and p-separator properties, minimizing maximum boundary costs in load balancing applications.

Contribution

It introduces a method to partition such graphs into nearly equal parts with boundary costs close to theoretical lower bounds, extending previous results to weighted and more general settings.

Findings

Partitioning achieves boundary costs proportional to (SUM c_e^p / k)^(1/p) plus maximum edge cost.

Partitioning can be computed efficiently, nearly as fast as finding a separator.

Bounds are tight up to a constant factor for various graph instances.

Abstract

Many load balancing problems that arise in scientific computing applications ask to partition a graph with weights on the vertices and costs on the edges into a given number of almost equally-weighted parts such that the maximum boundary cost over all parts is small. Here, this partitioning problem is considered for bounded-degree graphs G=(V,E) with edge costs c: E->R+ that have a p-separator theorem for some p>1, i.e., any (arbitrarily weighted) subgraph of G can be separated into two parts of roughly the same weight by removing a vertex set S such that the edges incident to S in the subgraph have total cost at most proportional to (SUM_e c^p_e)^(1/p), where the sum is over all edges e in the subgraph. We show for all positive integers k and weights w that the vertices of G can be partitioned into k parts such that the weight of each part differs from the average weight by less than MAX{w_v; v in V}, and the boundary edges of each part have cost at most proportional to (SUM_e c_e^p/k)^(1/p) + MAX_e c_e. The partition can be computed in time nearly proportional to the time for computing a separator S of G. Our upper bound on the boundary costs is shown to be tight up to a constant factor for infinitely many instances with a broad range of parameters. Previous results achieved this bound only if one has c=1, w=1, and one allows parts with weight exceeding the average by a constant fraction.