Characterize families without constant-factor approximation algorithms

Characterize the combinatorial feasibility families that necessarily admit no constant-factor approximation algorithm for the expected Top-k sum objective under independent nonnegative discrete item values.

Background

The paper establishes that no single polynomial-time algorithm achieves a constant-factor approximation for the expected Top-k sum objective across all membership families, already for k=1, using independent sets as a witness. It also provides a constant-factor transfer guarantee whenever deterministic max-sum admits an approximation, and therefore covers fixed-dimensional packing, cuts, and other families with suitable max-sum algorithms.

What remains unresolved is a per-family classification: identifying precisely which concrete combinatorial families inherently preclude any constant-factor approximation for expected Top-k sum. This question concerns the boundary between families such as independent sets, for which no constant factor is possible under standard complexity assumptions, and families admitting a constant-factor method through max-sum approximation.