Degree complexity for a modified pigeonhole principle
We consider a modification of the pigeonhole principle, M P H P, introduced by Goerdt in . M P H P is defined over n pigeons and log n holes, and more than one pigeon can go into a hole (according to some rules). Using a technique of Razborov  and simplified by Impagliazzo, Pudlák and Sgall , we prove that any Polynomial Calculus refutation of a set of polynomials encoding the M P H P, requires degree Ω(log n). We also prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus. Using this lemma, and a Resolution upper bound by Goerdt , we obtain that the degree lower bound is tight.
Our lower bound establishes the optimality of the tree-like Resolution simulation by the Polynomial Calculus given in .
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