Fast Balanced Partitioning Is Hard Even on Grids and Trees

Abstract Two kinds of approximation algorithms exist for the k-BAL-ANCED PARTITIONING problem: those that are fast but compute unsatis-fying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is necessary. 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 reduction framework which identifies some necessary 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 2D grid without holes. First we use the framework to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfying ratio. Then we consider solutions in which the sets may deviate from being equal-sized. Our framework is used on 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 the more stringent the limit on the set sizes is. These are the first bicriteria inapproximability results for the problem. 1 The Model and Setting We consider the k-BALANCED PARTITIONING problem in which the n ver-tices of a graph need to be partitioned into k sets of size at most dn/ke each. At the same time the cut size, which is the number of edges connecting ver-tices from different sets, needs to be minimised. This problem has many ap-plications including VLSI circuit design [4], image processing [26], computer vision [18], route planning [5], and divide-and-conquer algorithms [20]. In our case the motivation stems from parallel computations for finite element models (FEMs) [2]. In these a continuous domain of a physical model is discretised into a mesh of sub-domains (the elements). The mesh induce