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Separations above TFNP from Sherali-Adams Lower Bounds

Anna Gal, Noah Fleming, Deniz Imrek, Christophe Marciot

Abstract

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.

Separations above TFNP from Sherali-Adams Lower Bounds

Abstract

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) 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 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 -variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the setting, showing that the existence of a pseudo-expectation precludes a 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.
Paper Structure (20 sections, 20 theorems, 27 equations, 1 figure)

This paper contains 20 sections, 20 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Avoid.
  3. Linear Ordering Principle ( LOP).
  4. Strong Avoid.
  5. Least Number.
  6. Technical Highlights
  7. Related Works
  8. Preliminaries on Black-Box $\TF\Sigma_2$
  9. Proof Complexity and a Sufficient Condition for Separations
  10. Separating LOP from Strong Avoid
  11. The No-Minimal-Element Axioms
  12. A Warmup
  13. Normalization with Respect to Juntas
  14. The General Case
  15. A Criterion for Non-Reducibility of ${\text{\upshape\scshape LOP}}\xspace$
  16. ...and 5 more sections

Key Result

Theorem 1.1

${\text{\upshape\scshape LOP}}\xspace \not \in \sAvoid^{dt}$.

Figures (1)

  • Figure 1: Relationships of some $\TF\Sigma_2^{dt}$ classes. A black arrow from a class $A$ to a class $B$ means that $A\subseteq B$ (KleinbergKMP21). A dashed or dotted arrow from a class $A$ to a class $B$ means that $A\not\subseteq B$. The dashed separations are proved in this paper. KortenP24 proves that ${\text{\upshape\scshape StrongAvoid}}\xspace$ does not reduce to ${\text{\upshape\scshape Avoid}}\xspace$ and ${\text{\upshape\scshape LOP}}\xspace$. FlemingGRJLSY25 proves that ${\text{\upshape\scshape Avoid}}\xspace^{dt}$ does not contain all of $\TFNP^{dt}$. This implies that none of the other problems in the diagram reduces to ${\text{\upshape\scshape Avoid}}\xspace$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: FlemingIM25
  • Definition 3.1
  • Theorem 3.2: FlemingIM25
  • Definition 3.3
  • ...and 40 more