Type matching problems occur in a number of contexts, including library search, component composition, and inhabitation. We consider the intersection type matching problem under the standard notion of subtyping for intersection types: Given intersection types τ and σ, where σ is a constant type, does there exist a type substitution S such that S(τ) is a subtype of σ? We show that the matching problem is NP-complete. NP-hardness holds already for the restriction to atomic substitutions. The main contribution is an NP-algorithm which is engineered for efficiency by minimizing nondeterminism and running in Ptime on deterministic input problems. Our algorithm is based on a nondeterministic polynomial time normalization procedure for subtype constraint systems with intersection types. We have applied intersection type matching in optimizations of an inhabitation algorithm.