Resolution lower bounds for perfect matching principles

Abstract

For an arbitrary hypergraph H let PM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) must have size exp((/spl Omega/(/spl delta/(H)//spl lambda/(H)r(H)(log n(H))(r(H)+log n(H)))), where n(H) is the number of vertices, /spl delta/(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and /spl lambda/(H) is the maximal number of edges incident to two different vertices. For ordinary graphs G our general bound considerably simplifies to exp (/spl Omega/(/spl delta/(G)/(log n(G))/sup 2/))). As a direct corollary, every resolution proof of the functional onto a version of the pigeonhole principle onto - FPHP/sub n//sup m/ must have size exp (/spl Omega/(n/(log m)/sup 2/)) (which becomes exp (/spl Omega/(n/sup 1/3/)) when the number of pigeons m is unbounded). This in turn immediately implies an exp(/spl Omega/(t/n/sup 3/)) lower bound on the size of resolution proofs of the principle circuit/sub t/(f/sub n/) asserting that the circuit size of the Boolean function f/sub n/ in n variables is greater than t. In particular resolution does not possess efficient proofs of NP /spl subne/ P/poly. These results relativize, in a natural way, to more general principle M(U|H) asserting that H contains a matching covering all vertices in U /spl sube/ V(H).