Bridging the value–solution gap for clustering in MPC

Determine whether the gap between value estimation and computing approximate solutions for Euclidean (k,z)-Clustering in the Massively Parallel Computation model can be closed by developing fully-scalable algorithms that achieve solution-quality guarantees comparable to the O(1)-approximate value estimation achievable in O(1) rounds.

Background

The paper presents an MPC algorithm that estimates the optimal objective value of Euclidean (k,z)-Clustering within an O(1) factor in O(1) rounds, while the best solution algorithm provided achieves an O((log n / log log n)^z)-approximation in the same round regime.

This creates a separation between what can be achieved for the value of the objective and for constructing a concrete set of k centers under fully-scalable, constant-round MPC constraints. Closing this gap would align solution guarantees with value-estimation guarantees and represents a central challenge highlighted by the authors.