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DOI: 10.4230/LIPIcs.CCC.2026.37

Separations Above TFNP from Sherali-Adams Lower Bounds

Authors: Noah Fleming, Anna Gál, Deniz Imrek, and Christophe Marciot

Published in: LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026)

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. [Robert Kleinberg et al., 2021] observed that many of the natural problems considered so far in the second level of the total function polynomial hierarchy (TFΣ₂) 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 TFΣ₂ 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 [Fleming et al., 2025]. 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 Σ₂-variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the Σ₂ setting, showing that the existence of a Σ₂ pseudo-expectation precludes a Σ₂ 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.

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Noah Fleming, Anna Gál, Deniz Imrek, and Christophe Marciot. Separations Above TFNP from Sherali-Adams Lower Bounds. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 37:1-37:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fleming_et_al:LIPIcs.CCC.2026.37,
  author =	{Fleming, Noah and G\'{a}l, Anna and Imrek, Deniz and Marciot, Christophe},
  title =	{{Separations Above TFNP from Sherali-Adams Lower Bounds}},
  booktitle =	{41st Computational Complexity Conference (CCC 2026)},
  pages =	{37:1--37:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-437-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{383},
  editor =	{Moshkovitz, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.37},
  URN =		{urn:nbn:de:0030-drops-270797},
  doi =		{10.4230/LIPIcs.CCC.2026.37},
  annote =	{Keywords: TFNP, Range Avoidance, Linear Ordering Principle, Separation, Sherali-Adams, Pseudo-Expectation}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.28

Provably Total Functions in the Polynomial Hierarchy

Authors: Noah Fleming, Deniz Imrek, and Christophe Marciot

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

TFNP studies the complexity of total, verifiable search problems, and represents the first layer of the total function polynomial hierarchy (TFPH). Recently, problems in higher levels of the TFPH have gained significant attention, partly due to their close connection to circuit lower bounds. However, very little is known about the relationships between problems in levels of the hierarchy beyond TFNP. Connections to proof complexity have had an outsized impact on our understanding of the relationships between subclasses of TFNP in the black-box model. Subclasses are characterized by provability in certain proof systems, which has allowed for tools from proof complexity to be applied in order to separate TFNP problems. In this work we begin a systematic study of the relationship between subclasses of total search problems in the polynomial hierarchy and proof systems. We show that, akin to TFNP, reductions to a problem in TFΣ_d are equivalent to proofs of the formulas expressing the totality of the problems in some Σ_d-proof system. Having established this general correspondence, we examine important subclasses of TFPH. We show that reductions to the StrongAvoid problem are equivalent to proofs in a Σ₂-variant of the (unary) Sherali-Adams proof system. As well, we explore the TFPH classes which result from well-studied proof systems, introducing a number of new TFΣ₂ classes which characterize variants of DNF resolution, as well as TFΣ_d classes capturing levels of Σ_d-bounded-depth Frege.

Cite as

Noah Fleming, Deniz Imrek, and Christophe Marciot. Provably Total Functions in the Polynomial Hierarchy. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 28:1-28:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fleming_et_al:LIPIcs.CCC.2025.28,
  author =	{Fleming, Noah and Imrek, Deniz and Marciot, Christophe},
  title =	{{Provably Total Functions in the Polynomial Hierarchy}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{28:1--28:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.28},
  URN =		{urn:nbn:de:0030-drops-237223},
  doi =		{10.4230/LIPIcs.CCC.2025.28},
  annote =	{Keywords: TFNP, TFPH, Proof Complxity, Characterizations}
}

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Documents authored by Marciot, Christophe

Separations Above TFNP from Sherali-Adams Lower Bounds

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Provably Total Functions in the Polynomial Hierarchy

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