Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

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

Complexity and algorithms for convex network optimization and other nonlinear problems

“…One way which is particularly efficient for solving the scaled problem in step 1 is an adaptation of the successive shortest paths method (due to Jewell 1958;Iri 1960;Busacker and Gowen 1961 with some improvements by Edmonds and Karp 1972). We remark later about potential other approaches for solving the scaled problem.…”

“…It is interesting to note thatEdmonds and Karp (1972) used such idea of capacity scaling for the maximum flow problem that can be formulated as a minimum cost problem with b = 0. This network flow problem readily provides feasible integer solutions as the right hand sides are 0 and thus integers.…”

Deploying a massively multiplayer online game with a low-latency server infrastructure

“…More specifically, steps 3-5 involve solving up to n relaxed assignment problems and steps 6-8 entail no more than n constrained assignment problems. Assignment problems can be efficiently solved by polynomial-time algorithm [5]. While the algorithm still needs to solve a mixed-integer assignment problem in each back-tracking iteration, the problem is much easier to solve than Problem P1 because the network server nodes have been pre-selected.…”

My experiences as a student and researcher in OR during the 1960’s and 70’s

“…He wrote his lecture notes on polymatroids, terse but complete (Edmonds, 1970). He wrote the max flow paper with Karp (Edmonds and Karp, 1972) (my reference is to the later, shall we say formal, publication in the ACM), and he wrote the extended abstract on Blossom I and b-matching with me that dated back to 1967 (Edmonds and Johnson, 1970). He explained that the organizers felt he had enough pages already but let him have a few more for b-matching.…”