Characterize constant-factor max-sum families without a PTAS

Determine which combinatorial feasibility families that admit a constant-factor approximation algorithm for deterministic max-sum necessarily do not admit a polynomial-time approximation scheme for the expected Top-k sum objective.

Background

A constant-factor approximation for deterministic max-sum yields a constant-factor approximation for expected Top-k sum, but the paper proves that this hypothesis is insufficient for a PTAS in general. The family of graph cuts illustrates this separation: Max-Cut has a constant-factor approximation, while a PTAS for the stochastic expected-maximum case would contradict known hardness results.

The unresolved issue is to classify, on a per-family basis, those feasibility systems for which constant-factor max-sum approximability does not extend to a PTAS for expected Top-k sum. The desired classification would refine the paper’s uniform negative result beyond the cut witness.

References

A classification of which concrete F necessarily admit no constant-factor algorithm, or no PTAS, is open (Section~\ref{sec:concl}).

— A PTAS for Non-Adaptive Stochastic Top-$k$ Sum under General Combinatorial Constraints  (2609.03685 - Liu, 3 Sep 2026) in Remark 2.14, Section 2.3.3; Section 12, “Conclusion and Open Problems”

Which exact-sum families, outside packing, admit an EPTAS or an FPTAS? Cardinality for $k=1$ already has an EPTAS~\citep{segev2021,segev2024}.

— A PTAS for Non-Adaptive Stochastic Top-$k$ Sum under General Combinatorial Constraints  (2609.03685 - Liu, 3 Sep 2026) in Section 12, “Conclusion and Open Problems”