Dynamic matchings and quasidynamic fractional matchings. I

This paper presents and solves in polynomial time the dynamic matching problem, an integer programming problem which involves matchings in a time-expanded infinite network. The initial model is a finite directed graph G = (V, E) in which each edge has an associated real-valued weight and an integral distance. We wish to "match" vertices over an infinite horizon, and we permit vertex i in period p to be matched to vertex i in period r if and only if there is an edge e = (i, i) of E with distance r-p or else an edge e = (j, i) of E with distance p-r. Equivalently, we construct a "dynamic graph" in which there is an edge incident to vertex i-p and to vertex j-r in the above cases. The weight of this matched edge in the dynamic (time-expanded) graph is the weight of e. The dynamic matching problem is to determine a matching M in the dynamic graph such that M has a maximum long-run average weight per period. We show that the infinite horizon dynamic matching problem is linearly transformable to the finite horizon Q-matching problem, which is shown to be solvable in polynomial time in Part II of this paper.