Balanced Partitions of Trees and Applications

We study the problem of finding the minimum number of edges that, when cut, form a partition of the vertices into k sets of equal size. This is called the k-BALANCED PARTITIONING problem. The problem is known to be inapproximable within any finite factor on general graphs, while little is known about restricted graph classes. We show that the k-BALANCED PARTITIONING problem remains APX-hard even when restricted to unweighted tree instances with constant maximum degree. If instead the diameter of the tree is constant we prove that the problem is NP-hard to approximate within n (c) , for any constant c < 1. If vertex sets are allowed to deviate from being equal-sized by a factor of at most 1+epsilon, we show that solutions can be computed on weighted trees with cut cost no worse than the minimum attainable when requiring equal-sized sets. This result is then extended to general graphs via decompositions into trees and improves the previously best approximation ratio from O(log(1.5)(n)/epsilon (2)) [Andreev and Racke in Theory Comput. Syst. 39(6):929-939, 2006] to O(logn). This also settles the open problem of whether an algorithm exists for which the number of edges cut is independent of epsilon.