A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.A matching in G is a subset of its edges such that no two meet the same vertex. We describe an efficient algorithm for finding in a given graph a matching of maximum cardinality. This problem was posed and partly solved by C. Berge; see Sections 3.7 and 3.8.

1Jack Edmonds (Dece mbe r I , 1964) A matc hing in a graph C is a s ub set of e dges in C suc h that no two meet the sa me nod e in C. The co nvex polyhedron C is c harac te ri zed, wh e re th e e xtreme points of C co rres pond to th e matc hin gs in C. Wh e re eac h e dge of C car ri es a real num e ri cal we ight, a n e ffi c ie nt algorithm is describ ed for findin g a ma tc hin g in C with max imum we ight· s um .

No abstract

ABSTRACT. This paper presents new algorithms for the maximum flow problem, the Hitchcock t r a n s p o r t a t i o n problem, and the general minimum-cost flow problem. Upper bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms.First, the paper states the maximum flow problem, gives the Ford-Fulkerson labeling method for its solution, and points out t h a t an improper choice of flow augmenting paths can lead to severe computational difficulties. Then rules of choice t h a t avoid these difficulties are given. We show that, if each flow augmentation is made along an augmenting p a t h having a minimum number of arcs, then a maximum flow in an n-node network will be obtained after no more than ~(n a -n) augmentations; and then we show t h a t if each flow change is chosen to produce a maximum increase in the flow value then, provided the capacities are integral, a maximum flow will be determined within at most 1 + logM/(M--1) if(t, S) augmentations, wheref*(t, s) is the value of the maximum flow and M is the maximum number of arcs across a cut.Next a new algorithm is given for the minimum-cost flow problem, in which all shortest-path computations are performed on networks with all weights nonnegative. In particular, this algorithm solves the n X n assigmnent problem in O(n 3) steps. Following t h a t we explore a "scaling" technique for solving a minimum-cost flow problem by treating a sequence of derived problems with "scaled down" capacities. It is shown that, using this technique, the solution of a Iiitchcock transportation problem with m sources and n sinks, m ~ n, and maximum flow B, requires at most (n + 2) log2