What is in #P and what is not?

Ikenmeyer, Christian; Pak, Igor

doi:10.48550/arxiv.2204.13149

Cited by 3 publications

(4 citation statements)

References 128 publications

(158 reference statements)

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“…Third, in the heart of our proof of Theorem 1.3 in §4.4, we follow the complexity roadmap championed by Ikenmeyer, Panova and the second author in [IP22,IPP22]. Same for the heart of the proof of the Verification Lemma 4.6 in §8.4, which follows the approach in our companion paper [CP23].…”

Section: Fixing One Elementmentioning

confidence: 99%

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The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Shenfeld,

van Handel

2023

Acta Mathematica

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Describing the equality conditions of the Alexandrov-Fenchel inequality has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem, and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH20], which gave a geometric characterization of the equality conditions. The proof involves Stanley's order polytopes and employs poset theoretic technology.

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Section: Fixing One Elementmentioning

confidence: 99%

“…where F, F ′ are CNF Boolean formulas and P, P ′ are posets [CP23,IP22]. In other words, all three functions in ( ) do not have a combinatorial interpretation (unless PH collapses).…”

mentioning

confidence: 99%

The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Shenfeld,

van Handel

2023

Acta Mathematica

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“…Let GapP ≥0 denote the subset of nonnegative functions in GapP. Many interesting functions in algebraic combinatorics are known to be in GapP ≥0 , but conjectured to be in #P. See [IP22,Pak19] for many such functions arising from combinatorial inequalities. The most famous GapP ≥0 functions are the subject of of Stanley's survey [Sta00] on positivity problems in algebraic combinatorics, where he asked for positive combinatorial interpretations of the plethysm, Kronecker, and Schubert coefficients.…”

mentioning

confidence: 99%

“…The relativizing closure properties of #P have been characterized in [HVW95], which can be generalized to prove non-containment in #P w.r.t. an oracle in several settings, see [IP22].…”

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confidence: 99%

Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

Ikenmeyer,

Pak,

Panova

2023

International Mathematics Research Notices

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We prove that deciding the vanishing of the character of the symmetric group is $\textsf{C}_= \textsf{P}$-complete. We use this hardness result to prove that the absolute value and also the square of the character are not contained in $\textsf{#P}$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof, we conclude that deciding positivity of the character is $\textsf{PP}$-complete under many-one reductions, and hence $\textsf{PH}$-hard under Turing reductions.

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