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### Revision(s):

Revision #2 to TR21-021 | 27th July 2022 17:20

#### Perfect Matching in Random Graphs is as Hard as Tseitin

Revision #2
Authors: Per Austrin, Kilian Risse
Accepted on: 27th July 2022 17:20
Keywords:

Abstract:

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n/\log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lovász-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

Changes to previous version:

This version addresses several issues pointed out by the journal referees.

Revision #1 to TR21-021 | 25th October 2021 16:58

#### Perfect Matching in Random Graphs is as Hard as Tseitin

Revision #1
Authors: Per Austrin, Kilian Risse
Accepted on: 25th October 2021 16:58
Keywords:

Abstract:

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n/\log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lovász-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

Changes to previous version:

Incorporated various reviewer comments which hopefully improved the exposition.

### Paper:

TR21-021 | 18th February 2021 18:12

#### Average-Case Perfect Matching Lower Bounds from Hardness of Tseitin Formulas

TR21-021
Authors: Per Austrin, Kilian Risse
Publication: 20th February 2021 20:26
We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n/\log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lovász-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.