Multiply Balanced k −Partitioning

Abstract

The problem of partitioning an edge-capacitated graph on n vertices into k balanced parts has been amply researched. Motivated by applications such as load balancing in distributed systems and market segmentation in social networks, we propose a new variant of the problem, called Multiply Balanced k Partitioning, where the vertex-partition must be balanced under d vertex-weight functions simultaneously.

We design bicriteria approximation algorithms for this problem, i.e., they partition the vertices into up to k parts that are nearly balanced simultaneously for all weight functions, and their approximation factor for the capacity of cut edges matches the bounds known for a single weight function times d. For the case where d = 2, for vertex weights that are integers bounded by a polynomial in n and any fixed ε > 0, we obtain a $(2+\epsilon,\, O(\sqrt{\log n \log k}))$-bicriteria approximation, namely, we partition the graph into parts whose weight is at most 2 + ε times that of a perfectly balanced part (simultaneously for both weight functions), and whose cut capacity is $O(\sqrt{\log n \log k})\cdot$ OPT. For unbounded (exponential) vertex weights, we achieve approximation $(3,\ O(\log n))$.

Our algorithm generalizes to d weight functions as follows: For vertex weights that are integers bounded by a polynomial in n and any fixed ε > 0, we obtain a $(2d +\epsilon,\, O(d\sqrt{\log n \log k}))$-bicriteria approximation. For unbounded (exponential) vertex weights, we achieve approximation $(2d+ 1,\ O(d\log n))$.