Perfect Matching in Random Graphs is as Hard as Tseitin

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 Lov ász-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.