Separations above TFNP from Sherali-Adams Lower Bounds

Unlike in TFNP, for which there is an abundance of problems capturing natural existence principles which are incomparable (in the black-box setting), Kleinberg et al. [KKMP21] observed that many of the natural problems considered so far in the second level of the total function polynomial hierarchy (TF$Σ_2$) reduce to the Strong Avoid problem. In this work, we prove that the Linear Ordering Principle does not reduce to Strong Avoid in the black-box setting, exhibiting the first TF$Σ_2$ problem that lies outside of the class of problems reducible to Strong Avoid. The proof of our separation exploits a connection between total search problems in the polynomial hierarchy and proof complexity, recently developed by Fleming, Imrek, and Marciot [FIM25]. In particular, this implies that to show our separation, it suffices to show that there is no small proof of the Linear Ordering Principle in a $Σ_2$-variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the $Σ_2$ setting, showing that the existence of a $Σ_2$ pseudo-expectation precludes a $Σ_2$ Sherali-Adams proof. The main technical challenge is in proving the existence of such a pseudo-expectation, we manage to do so by solving a combinatorial covering problem about permutations. We also show that the extended pseudo-expectation bound implies that the Linear Ordering Principle cannot be reduced to any problem admitting a low-degree Sherali-Adams refutation.

💡 Research Summary

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The paper investigates the landscape of total search problems at the second level of the polynomial hierarchy, denoted TF Σ₂, and establishes the first natural problem in this class that does not reduce to the Strong Avoid problem in the black‑box (decision‑tree) setting. The problem in question is the Linear Ordering Principle (LOP), a classic combinatorial principle that asks for either a minimal element or a violation of transitivity in a given binary relation on a set of size 2ⁿ.

The authors build on a recent connection between black‑box reductions for TF Σ₂ problems and proof complexity introduced by Fleming, Imrek, and Marciot (2025). Specifically, they show that a TF Σ₂ problem R reduces to Strong Avoid with polylogarithmic decision‑tree complexity if and only if the unsatisfiable DNF formula encoding the totality of R admits an efficient Σ₂‑variant of the Sherali‑Adams (SA) proof system, called Σ₂‑uSA. In Σ₂‑uSA, besides the usual SA derivations, one is allowed a Σ₂‑weakening step that replaces a DNF by a collection of DNFs that together imply the original formula, mirroring the ∃∀ structure of Σ₂ statements.

To prove that LOP does not have such a proof, the authors extend the pseudo‑expectation technique, a standard tool for SA lower bounds, to the Σ₂ setting. A pseudo‑expectation is a linear functional that behaves like expectation over a probability distribution on assignments, but only on low‑degree polynomials. The existence of a degree‑d pseudo‑expectation for a set of constraints rules out any SA proof of degree ≤ d. The Σ₂‑pseudo‑expectation defined here is a family of pseudo‑expectations, one for each possible Σ₂‑weakening of the original constraints, and it must simultaneously satisfy all of them.

Constructing such a family is technically demanding. The authors reduce the construction to a combinatorial covering problem on permutations. Let Ord be the set of all total orders on {1,…,n} and Ord*₁ the subset of orders where the element 1 does not appear first. For a fixed d ≤ n, consider all subsets S ⊆