Dual coordinate step methods for linear network flow problems

Abstract

We review a class of recently-proposed linear-cost network flow methods which are amenable to distributed implementation. All the methods in the class use the notion of e-complementary slackness, and most do not explicitly manipulate any "global" objects such as paths, trees, or cuts. Interestingly, these methods have stimulated a large number of new serial computational complexity results. We develop the basic theory of these methods and present two specific methods, the e-relaxation algorithm for the minimum-cost flow problem, and the auction algorithm for the assignment problem. We show how to implement these methods with serial complexities of O(N 3 log NC) and O(NA log NC), respectively. We also discuss practical implementation issues and computational experience to date. Finally, we show how to implement e-relaxation in a completely asynchronous, "chaotic" environment in which some processors compute faster than others, some processors communicate faster than others, and there can be arbitrarily large communication delays.

Auction algorithms for network flow problems: A tutorial introduction

“…For integer data, it can be shown that the worst-case running time of the auction algorithm using scaling and appropriate data structures is O nA log(nC) ; see [BeE88], [BeT89]. Based on experiments, the running time of the algorithm for randomly generated problems seems to grow proportionally to something like A log n or A log n log(nC).…”

“…The complexity of the scaled version was first studied in [Gol87], where particularly favorable polynomial running time estimates were derived; see [BeE87], [BeE88], [GoT90] for additional results along these lines.…”

“…Computational studies [BCT91], [BeE88] have shown that it is often substantially faster than its closest competitors, particularly for sparse problems. The speedup factor typically increases with the problem size.…”

“…However, the -relaxation method is well suited for parallel computation, so in some parallel computing environments it may be faster than its closest competitors. The method admits a totally asynchronous implementation, as shown in the original paper [Ber86a]; see also [BeE88], [BeT89].…”

“…This algorithm was further developed in [Ber85], [Ber88], and [BeE88]. The motivation was to solve the problem by using parallelism in a natural way.…”