Clustering with Penalty for Joint Occurrence of Objects: Computational Aspects

TL;DR

This paper investigates the computational complexity of a clustering method that penalizes joint occurrence of objects, proving NP-hardness and proposing heuristic algorithms to improve practical solution finding.

Contribution

The paper proves the NP-hardness of the clustering problem and introduces enhanced genetic algorithm techniques for better computational performance.

Findings

NP-hardness of the clustering problem established

Proposed genetic algorithm with heuristics improves solution efficiency

Simulation results show significant performance enhancements

Abstract

The method of Hol\'y, Sokol and \v{C}ern\'y (Applied Soft Computing, 2017, Vol. 60, p. 752-762) clusters objects based on their incidence in a large number of given sets. The idea is to minimize the occurrence of multiple objects from the same cluster in the same set. In the current paper, we study computational aspects of the method. First, we prove that the problem of finding the optimal clustering is NP-hard. Second, to numerically find a suitable clustering, we propose to use the genetic algorithm augmented by a renumbering procedure, a fast task-specific local search heuristic and an initial solution based on a simplified model. Third, in a simulation study, we demonstrate that our improvements of the standard genetic algorithm significantly enhance its computational performance.