An incremental exact algorithm for the hyper-rectangular clustering problem with axis-parallel clusters

TL;DR

This paper introduces an adaptive incremental exact algorithm for the hyper-rectangular clustering problem with axis-parallel clusters, improving the ability to solve larger instances optimally compared to previous methods.

Contribution

The paper presents a novel adaptive incremental strategy that efficiently solves larger instances of the hyper-rectangular clustering problem by building on small instance solutions.

Findings

Successfully solves larger instances to optimality

Outperforms existing methods in computational experiments

Proves the optimality of solutions when covering all points

Abstract

We address the problem of clustering a set of points in $\mathbb{R}^d$ with axis-parallel clusters. Previous exact approaches to this problem are mostly based on integer programming formulations and can only solve to optimality instances of small size. In this work we propose an adaptive exact strategy which takes advantage of the capacity to solve small instances to optimality of previous approaches. Our algorithm starts by solving an instance with a small subset of points and iteratively adds more points if these are not covered by the obtained solution. We prove that as soon as a solution covers the whole set of point from the instance, then the solution is actually an optimal solution for the original problem. We compare the efficiency of the new method against the existing ones with an exhaustive computational experimentation in which we show that the new approach is able to solve to optimality instances of higher orders of magnitude.