Efficient solutions for weight-balanced partitioning problems

TL;DR

This paper introduces polynomial-time algorithms for a broad class of weight-balanced clustering problems, with applications in land consolidation, by leveraging fixed dimensions and finite domain constraints.

Contribution

It proves polynomial solvability for a class of clustering problems with complex balancing constraints, extending the scope of efficient solutions in this domain.

Findings

Polynomial-time solvability for fixed dimensions and clusters

Applicable to land consolidation with efficient algorithms

Handles convex objective functions in clustering

Abstract

We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to different features and the clusters adhere to given lower and upper bounds on the total weight of their points with respect to each of these features. Further the weight-contribution of a vector to a cluster can depend on the cluster it is assigned to. Our interest in these types of clustering problems is motivated by an application in land consolidation where the ability to perform this kind of balancing is crucial. Our framework maximizes an objective function that is convex in the summed-up utility of the items in each cluster. Despite hardness of convex maximization and many related problems, for fixed dimension and number of clusters, we are able to show that our clustering model is solvable in time polynomial in the number of items if the weight-balancing restrictions are defined using vectors from a fixed, finite domain. We conclude our discussion with a new, efficient model and algorithm for land consolidation.