Many to Many Matching with Demands and Capacities

Let A and B be two finite sets of points with total cardinality n, the many to many point matching with demands and capacities matches each point ai ∈ A to at least αi and at most α ′ i points in B, and each point bj ∈ B to at least βj and at most β ′ j points in A for all 1 ≤ i ≤ s and 1 ≤ j ≤ t. In this paper, we present an upper bound for this problem using our new polynomial time algorithm.