Refuting Perfect Matchings in Spectral Expanders is Hard
Abstract
This work studies the complexity of refuting the existence of a perfect matching in spectral expanders with an odd number of vertices, in the Polynomial Calculus (PC) and Sum of Squares (SoS) proof system. Austrin and Risse [SODA, 2021] showed that refuting perfect matchings in sparse d-regular random graphs, in the above proof systems, with high probability requires proofs with degree (n/ n). We extend their result by showing the same lower bound holds for all d-regular graphs with a mild spectral gap.
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