EKM: An exact, polynomial-time algorithm for the $K$-medoids problem

TL;DR

This paper introduces EKM, a polynomial-time algorithm that guarantees an exact solution to the $K$-medoids clustering problem, outperforming traditional exponential-time methods on real-world datasets.

Contribution

The paper presents EKM, the first rigorously-proven polynomial-time algorithm for the $K$-medoids problem that finds exact solutions, using formal derivation techniques.

Findings

EKM solves the $K$-medoids problem exactly in polynomial time.

EKM outperforms approximate methods on real-world datasets.

EKM's runtime matches its worst-case complexity, outperforming exponential-time solvers.

Abstract

The $K$-medoids problem is a challenging combinatorial clustering task, widely used in data analysis applications. While numerous algorithms have been proposed to solve this problem, none of these are able to obtain an exact (globally optimal) solution for the problem in polynomial time. In this paper, we present EKM: a novel algorithm for solving this problem exactly with worst-case $O\left(N^{K+1}\right)$ time complexity. EKM is developed according to recent advances in transformational programming and combinatorial generation, using formal program derivation steps. The derived algorithm is provably correct by construction. We demonstrate the effectiveness of our algorithm by comparing it against various approximate methods on numerous real-world datasets. We show that the wall-clock run time of our algorithm matches the worst-case time complexity analysis on synthetic datasets, clearly outperforming the exponential time complexity of benchmark branch-and-bound based MIP solvers. To our knowledge, this is the first, rigorously-proven polynomial time, practical algorithm for this ubiquitous problem.