Coloring for dispersion: A polynomial-time algorithm for cardinality-constrained 2-anticlustering

TL;DR

This paper presents a polynomial-time algorithm for the 2-maximum dispersion problem with cardinality constraints, solving an open problem and enabling efficient large-scale data processing.

Contribution

It introduces a polynomial-time solution for 2-MDCC by transforming it into a restricted subset sum problem, outperforming previous methods.

Findings

The algorithm solves large datasets in less than a second.

It outperforms previous integer linear programming solutions by several orders of magnitude.

The approach confirms polynomial solvability for the 2-anticlustering problem.

Abstract

The $k$-Maximum Dispersion Problem with Cardinality Constraints ($k$-MDCC) asks for a partition of a given item set with pairwise dissimilarities into $k$ cardinality-constrained groups such that the minimum pairwise intra-group dissimilarity, which is also known as the dispersion, is maximized. The problem arises in the context of anticlustering, where the goal is to create maximally heterogeneous groups of items with applications in psychological research, bioinformatics, and data science. It is known that $k$-MDCC is NP-hard for $k \geq 3$ but it has been an open question whether it can be solved in polynomial time for $k = 2$. We give a positive answer to this question by showing that $2$-MDCC can be solved by a quadratic number of cardinality-constrained 2-coloring problem instances ($2$-COLCC). We solve these instances by transforming them into a restricted class of subset sum instances. Although subset sum is NP-complete in general, for this restricted class the input values are bounded, ensuring that the pseudopolynomial dynamic programming algorithm runs in polynomial time. As a consequence, we obtain a polynomial-time algorithm for $2$-MDCC. We demonstrate that a publicly available open-source implementation of our new algorithm outperforms the previous integer linear programming solution by several orders of magnitude so that even large datasets ($n = 10{,}000$) can be processed in less than a second.