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 Lov\'asz-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.

翻译：我们研究在奇数的脊椎上,一个稀有的随机常规图表并不完全匹配,而且每个顶端都存在相关问题,需要经过一定的预设次数才能匹配。 我们发现,这需要多式微积分系统(特征领域为$ne 2, 和Sum-quares验证系统)中的美元/ omega(n /\log n) $, 以及界限深度的Frege校准系统中的指数大小。 Razborov 询问Lov\'asz-Schrijver 验证系统是否需要 $n ⁇ delta$ 圆来反驳这些公式, 以 $\ delta > 0 。 最坏的情况是将这些公式平均减少, 依靠可能具有独立利益的表层嵌入质。