Fast balanced partitioning is hard even on grids and trees

Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfactory approximation ratios, and those that guarantee high quality ratios but are slow. In this article we prove that this tradeoff between running time and solution quality is unavoidable. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a general reduction which identifies some sufficient conditions on the considered graph class in order to prove the hardness of the problem. We focus on two combinatorially simple but very different classes, namely trees and solid grid graphs. The latter are finite connected subgraphs of the infinite two-dimensional grid without holes. We apply the reduction to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfactory ratio. We also consider solutions in which the sets may deviate from being equal-sized. Our reduction is applied to grids and trees to prove that no fully polynomial time algorithm exists that computes solutions in which the sets are arbitrarily close to equal-sized. This is true even if the number of edges cut is allowed to increase when the limit on the set sizes decreases. These are the first bicriteria inapproximability results for the k-BALANCED PARTITIONING problem.