Check, Please: Verifiably Fair Clustering

Popular centroid-based clustering methods are typically optimized for global objectives, and may fail to adequately represent large groups of datapoints. Thus, one needs proportionality notions suited for metric settings. Ideally, such notions should admit polynomial-time algorithms for (a) finding proportional outcomes, and (b) checking if a given outcome is proportional; the latter enables evaluation of traditional algorithms without proportionality guarantees (e.g., $k$-means). A promising approach imports proportionality notions from multiwinner voting with approval ballots. In particular, mPJR, the metric version of the well-known Proportional Justified Representation (PJR) axiom, satisfies (a), but whether it satisfies (b) was open. In this work, we study the computational complexity of auditing proportional representation in clustering. In the approval setting, PJR is coNP-complete to verify; however, it admits a strengthening PJR+, which satisfies (a) and (b). We show these results translate to the metric setting: mPJR is coNP-complete to verify, we define mPJR+, a metric analog of PJR+, and argue mPJR+ satisfies (a) and (b). However, auditing mPJR+ relies on repeated submodular minimization, rendering it impractical at scale, and a natural combinatorial approach is infeasible. As a partial remedy, we propose an mPJR+ verification algorithm exponential in $k$ but quasilinear in the number of datapoints. Motivated by these hardness results, we introduce DC-mPJR+: a proportionality concept offering representation guarantees to a restricted set of coalitions around unselected centers, admitting an $O(mn \log n + mnk)$ verification algorithm. DC-mPJR+ outcomes can be computed efficiently, and any $\gamma$-DC-mPJR+ solution satisfies $(\gamma + 2)$-mPJR+.