Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

v
f(u, v) -- ~ f(v, u) = O,
v
where each sum is over every v for which the s u m m a n d is defined. For each arc (u, v) of N, f(u, v) represents the a m o u n t of flow in arc (u, v), and also represents the net a m o u n t of flow from v to u in the rest of the network
Theoretical Imic Efficiency for Network Flow Problems
JACK EDMONDS
University of Waterloo, Waterloo, Ontario, Canada
AND
RICHARD M. K A R P
249
1.1. THE LABELING METHOD. A network N is a finiteset {u, v, - • • } called the nodes and a subset of the ordered pairs (u, v), u # v, called the arcs. N e t w o r k N has a special return arc (t, s). N o d e s is called the source in N and node t is called the sink in N. T h e set of all arcs of N, except (t, s), we denote b y A. For each (u, v) ~ A there is given a n u m b e r c(u, v) > 0 called the capacity of arc (u, v). A nonnegative function f(u, v), ranging over all arcs (u, v) of N, is called a flow in N if (i) for every (u, v) C A, f(u, v) < c(u, v); and (ii) for every node u,
"N (u, v ) . "
The m a x i m u m network flow problem is to find a flow f in N such that f(t, s), the net a m o u n t of flow in N - (t, s) from s to t, is maximum. Let u~, u2, • • • , u , be a sequence of distinct nodes such that, for each i = l, 2, • .. , p - 1, either (u~, u~+~) or (u~+~, u~) is an arc. Singling out, for each i, one of these possibilities, we call the resulting sequence of arcs a path from Ul to Up. Arcs (u~, u~+~) t h a t belong to the p a t h are called forward ares of the p a t h ; the other arcs of the p a t h are called reverse arcs. Relative to a n y given flow f in N, a (flow) augme~ti~tg path is a p a t h from s to t such that :
Jc~urnal of the Associationfor Computing Machinery, Vol. 19, No. 2, Apri| 1972. pp. 248-264.
Theoretical Improvements in Algorithmic E~ciency for Network Flow Problems 1. The Maximum Flow Problem
University of California, Berkeley, California
ABSTRACT. This paper presents new algorithms for t h e m a x i m u m flow problem, the Hitchcock t r a n s p o r t a t i o n problem, and t h e general m i n i m u m - c o s t flow problem. U p p e r bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on t h e numbers of steps required by earlier algorithms. First, the paper states the m a x i m u m flow problem, gives the F o r d - F u l k e r s o n labeling method for its solution, and points out t h a t an improper choice of flow a u g m e n t i n g p a t h s can lead to severe c o m p u t a t i o n a l difficulties. T h e n rules of choice t h a t avoid these difficulties are given. We show t h a t , if each flow a u g m e n t a t i o n is made along an a u g m e n t i n g p a t h h a v i n g a minimum n u m b e r of arcs, t h e n a m a x i m u m flow in an n-node network will be o b t a i n e d a f t e r no more t h a n ~(n a - n) a u g m e n t a t i o n s ; and t h e n we show t h a t if each flow change is chosen to produce a m a x i m u m increase in the flow value then, provided the capacities are integral, a m a x i m u m flow will be d e t e r m i n e d within at most 1 + logM/(M--1) if(t, S) a u g m e n t a t i o n s , wheref*(t, s) is the value of the maximum flow and M is the m a x i m u m n u m b e r of arcs across a cut. Next a new algorithm is given for the m i n i m u m - c o s t flow problem, in which all s h o r t e s t - p a t h c o m p u t a t i o n s are performed on networks with all weights nonnegative. In particular, this a l g o r i t h m solves the n X n assigmnent problem in O(n3) steps. Following t h a t we explore a " s c a l i n g " technique for solving a minimum-cost flow problem by t r e a t i n g a sequence of derived problems w i t h "scaled d o w n " capacities. It is shown t h a t , using this technique, the solution of a I i i t c h c o c k t r a n s p o r t a t i o n problem w i t h m sources and n sinks, m ~ n, and m a x i m u m flow B, requires at most (n + 2) log2 (B/n) flow a u g m e n t a t i o n s . Similar results are also given for the general minimum-cost flow problem. An a b s t r a c t s t a t i n g the main results of the present paper was presented at the Calgary I n t e r n a t i o n a l Conference on C o m b i n a t o r i a l Structures and T h e i r Applications, J u n e 1969. In a paper b y l)inic (1970) a result closely related to the main result of Section 1.2 is obtained. Dinic shows t h a t , in a network with n nodes and p arcs, a m a x i m u m flow can be computed in 0 (n2p) primitive operations b y an algorithm which a u g m e n t s along s h o r t e s t augmenting paths. KEY WOl¢l)S AND PHP~ASES: network flows, t r a n s p o r t a t i o n problem, analysis of algorithms CR CATEGOI{.IES: 5.3, 5.4, 8.3 Copyright © 1972, Association for C o m p u t i n g Machinery, Inc. General permission to republish, b u t not for profit, all or p a r t of this m a t e r i a l is granted, provided t h a t reference is made to this publication, to its date of issue, and to the fact t h a t r e p r i n t i n g privileges were granted by permission of the Association for C o m p u t i n g Machinery. Authors' addresses : J. Edmonds, D e p a r t m e n t of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, C a n a d a ; R. M. Karp, College of Engineering, Operations Research Center, U n i v e r s i t y of California, Berkeley, CA 94720; the l a t t e r a u t h o r ' s research has been partially s u p p o r t e d by the N a t i o n a l Science F o u n d a t i o n raider G r a n t GP-15473 with the U n i v e r s i t y of California.