Perfect Matching in Random Graphs is as Hard as Tseitin

01/26/2022

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by   Per Austrin, et al.

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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 Ω(n / log n) in the Polynomial Calculus (over fields of characteristic 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^δ rounds to refute these formulas for some δ > 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.