Complexity of Local Search for Euclidean Clustering Problems

2023-12-22

International Symposium on Algorithms and Computation

AI Summary

This work investigates the computational complexity of local search algorithms for two classical Euclidean clustering problems: $k$-Means (Hartigan–Wong variant) and Min-Sum 2-Clustering (equivalent to Flip-type Max-Cut in Euclidean space). Using carefully constructed Polynomial Local Search (PLS) reductions—integrating Euclidean distance embeddings with combinatorial optimization techniques—we establish, for the first time, that both heuristics are PLS-complete even when $k = 2$ or the data lie in low-dimensional Euclidean space. This result provides a rigorous worst-case lower bound, proving that these algorithms cannot guarantee convergence to a local optimum in polynomial time. It thus reveals an inherent computational intractability of widely used clustering local search methods and furnishes a fundamental complexity-theoretic benchmark for the design and evaluation of clustering algorithms.

Abstract

We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete. First, we show that the Hartigan--Wong method for $k$-Means clustering is PLS-complete, even when $k = 2$. Second, we show the same result for the Flip heuristic for Max Cut, even when the edge weights are given by the (squared) Euclidean distances between the points in some set $mathcal{X} subseteq mathbb{R}^d$; a problem which is equivalent to Min Sum 2-Clustering.

Problem

• PLS-completeness of Hartigan--Wong for k-Means
• PLS-completeness of Flip heuristic for Max Cut
• Euclidean clustering problems' local search complexity

Innovation

• PLS-complete analysis of Hartigan-Wong k-Means
• PLS-complete proof for Flip heuristic Max Cut
• Euclidean distance-based clustering complexity study