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 vertexweights that are integers bounded by a polynomial in n and any fixed ∈ > 0, we obtain a (2+∈, O( √ log n 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( √ log n log k) 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 ∈ > 0, we obtain a (2d + ∈, O(√ log n log k))-bicriteria approximation. For unbounded (exponential) vertex weights, we achieve approximation (2d + 1, O(d log n)).