The computational complexity of some explainable clustering problems

TL;DR

This paper investigates the computational difficulty of explainable clustering problems using axis-aligned decision trees, showing some are NP-hard while others are solvable efficiently.

Contribution

It establishes the complexity status of several explainable clustering problems, identifying which can be optimized in polynomial time and which are computationally hard.

Findings

k-means, k-medians, and k-centers are NP-hard to optimize

Spacing cost function can be optimized in polynomial time

Clarifies the computational landscape of explainable clustering problems

Abstract

We study the computational complexity of some explainable clustering problems in the framework proposed by [Dasgupta et al., ICML 2020], where explainability is achieved via axis-aligned decision trees. We consider the $k$-means, $k$-medians, $k$-centers and the spacing cost functions. We prove that the first three are hard to optimize while the latter can be optimized in polynomial time.