Generalized Planar Matching

Abstract In this paper, we prove that maximum planar H-matching (the problem of determining the maximum number of node-disjoint copies of the fixed graph H contained in a variable planar graph G) is NP-complete for any connected planar graph H with three or more nodes. We also show that perfect planar H-matching is NP-complete for any connected outerplanar graph H with three or more nodes. The results generalize and unify several special-case results proved in the literature. The techniques can also be applied to determine the complexity of several problems, including the optimal tile salvage problem from wafer-scale integration and the classic dots and boxes game. Although we prove that the optimal tile salvage problem and others like it are NP-complete, we also describe provably good approximation algorithms that are suitable for practical applications.