Resolution lower bounds for perfect matching principles

• Authors: Alexander A. Razborov
• Venue: Journal of Computer and System Sciences - Special issue on computational complexity 2002
• Year: 2004

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(Ω(δ(H)/λ(H)r(H)(log n(H))(r(H) + log n(H)))), where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and λ(H) is the maximal number of edges incident to two different vertices.For ordinary graphs G our general bound considerably simplifies to exp(Ω(δ(G)/(log n(G))2)) (implying an exp(Ω(δ(G)1/3)) lower bound that depends on the minimal degree only). As a direct corollary, every resolution proof of the functional onto version of the pigeonhole principle onto - FPHPmn must have size exp(Ω(n/(log m)2)) (which becomes exp(ω(n1/3)) when the number of pigeons m is unbounded). This in turn immediately implies an exp(Ω(t/n3)) lower bound on the size of resolution proofs of the principle asserting that the circuit size of the Boolean function fn in n variables is greater than t. In particular, Resolution does not possess efficient proofs of NP ⊈ P/poly.These results relativize, in a natural way, to a more general principle M(U|H) asserting that H contains a matching covering all vertices in U ⊆ V(H).