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Small Set Expansion (SSE) Hypothesis

Establish the Small Set Expansion hypothesis by proving or refuting that, for any eta > 0, there exists a constant tau in (0,1) such that no polynomial-time algorithm can distinguish between the following two cases for a D-regular graph G = (V, E) on n vertices: (i) YES case—there exists a subset S ⊂ V of size |S| = tau n whose induced subgraph is dense, i.e., the number of edges leaving S satisfies |E(S, V \ S)| ≤ eta · D · |S|; and (ii) NO case—every subset S ⊂ V with |S| ≤ 2 tau n has near-complete expansion, i.e., |E(S, V \ S)| ≥ (1 − eta) · D · |S|.

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Background

The paper’s hardness result for proper learning with slack in coverage (Theorem 6.2) is conditioned on the Small Set Expansion (SSE) hypothesis. SSE is closely related to the Unique Games Conjecture and posits that distinguishing dense induced subgraphs from small-set expanders is computationally intractable.

Settling SSE (either by proof or refutation) would clarify the complexity landscape underlying the paper’s inapproximability results for learning minimum-volume confidence sets, particularly in high dimensions when coverage constraints are relaxed by a constant factor.

References

Conjecture [SSE hypothesis of Raghavendra and Steurer] For any eta > 0, there is a constant tau in (0,1) such that there is no polynomial time algorithm to distinguish between the following two cases given a graph G=(V,E) on n vertices with degree D: YES: Some subset S subseteq V with |S| = tau n satisfies that the induced subgraph on S is dense i.e., the number of edges going out of S is |E(S,V \ S)| ≤ eta D |S| edges. NO: Any set S subseteq V with |S| ≤ 2 tau n has most of the edges incident on it going outside i.e., |E(S,V \ S)| ≥ (1 − eta) |S| D.

— Computing High-dimensional Confidence Sets for Arbitrary Distributions (2504.02723 - Gao et al., 3 Apr 2025) in Section 6.2, Conjecture (Computational Intractability even with Slack in Coverage)