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+ ε , 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 )/ ε 2 ) [Andreev and Räcke in Theory Comput. Syst. 39(6):929–939, 2006 ] to O (log n ). This also settles the open problem of whether an algorithm exists for which the number of edges cut is independent of ε .