Multiply Balanced k -Partitioning

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 > 0, we obtain a (2 + , O( √ logn log k))-bicriteria approximation, namely, we partition the graph into parts whose weight is at most 2+ times that of a perfectly balanced part (simultaneously for both weight functions), and whose cut capacity is O( √ logn log k) · OPT. For unbounded (exponential) vertex weights, we achieve approximation (3, O(logn)). Our algorithm generalizes to d weight functions as follows: For vertex weights that are integers bounded by a polynomial in n and any fixed > 0, we obtain a (2d + , O(d √ logn log k))-bicriteria approximation. For unbounded (exponential) vertex weights, we achieve approximation (2d+ 1, O(d logn)).