A Note on Polynomial-time Solvability for Edge Packing on Graphs

Abstract

In this paper, we consider a generalized matching problem, the so-called edge packing problem. Given a graph G such that each vertex v is associated with a positive integer w(v), an edge packing F with respect to these integers (abbreviated as edge packing) is a collection of edges (repetition is allowed) of G such that each vertex (counting multiplicities) v belongs to at most w(v) members of F. When all vertices are associated with 1's in G, then edge packings are the usual matchings. The maximum edge packing problem asks for an edge packing containing edges as many as possible. We design an algorithm that solves this problem in polynomial-time, showing that this problem is polynomial-time solvable. Furthermore, our algorithm relies heavily on algorithms that solve b-matching in strongly polynomial time, showing that any strongly polynomial-time algorithm for b-matching provides a strongly polynomial-time algorithm for the maximum edge packing problem.