Bi-criteria set-packing hardness conjecture

Prove or refute the conjecture that the following bi-criteria set-packing problem is computationally hard: given a family of subsets of {1,…,m}, construct a sub-family in which every element appears in at most C sets and whose cardinality is within factor C of the largest pairwise-disjoint sub-family.

Background

To understand constant-factor augmentation at the single-block level, the authors reduce the question to a bi-criteria set-packing problem. If this bi-criteria problem is indeed computationally hard, it would argue against the feasibility of constant-factor guarantees using constant slackness and extension.

This conjecture targets a foundational combinatorial barrier that, if true, would limit algorithmic approaches to block-level packing under mild capacity relaxations.

References

At the single-block level this question reduces to a bi-criteria set-packing task: given a family of subsets of {1,\dots,m}, choose a sub-family in which every element appears in at most~C sets, yet whose size is within factor~C of the largest pairwise-disjoint sub-family. We conjecture this problem is computationally hard, which would argue against a positive answer.

— Online Block Packing (2507.12357 - Eliezer et al., 16 Jul 2025) in Subsubsection “Constant guarantees with constant augmentation,” Open Problems and Future Work