Computational complexity of optimal weighted selection

Determine the computational complexity of computing an optimal weighted selection for a given linear-regression dataset under a prescribed point budget, including whether the optimum can be computed efficiently.

Background

The paper provides constructive polynomial-time selection procedures for all of the exact upper bounds it proves, including the budgets $2d-1$ in arbitrary dimension, $4$ in dimension $3$, and $5$ in dimension $4$. These procedures are not claimed to optimize the selection objective for an arbitrary input dataset.

The unresolved computational question concerns finding the globally optimal weighted selection itself, rather than merely producing a selection with the paper’s proved worst-case guarantee.

References

Second, our selections are constructive but not optimized; the complexity of computing an optimal weighted selection for a given dataset is open.

— Exact Risk Ratios for Weighted Data Selection in Linear Regression  (2608.28007 - Zhang, 28 Aug 2026) in Section 6, Discussion and open problems, paragraph “Further questions”