Dual coordinate step methods for linear network flow problems

A class of recently-proposed linear-cost network flow methods which are amenable to distributed implementation using the notion ofε-complementary slackness is reviewed, and two specific methods, theε-relaxation algorithm for the minimum-cost flow problem, and the auction algorithms for the assignment problem are presented.

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 ofε-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 newserial computational complexity results. We develop the basic theory of these methods and present two specific methods, theε-relaxation algorithm for the minimum-cost flow problem, and theauction algorithm for the assignment problem. We show how to implement these methods with serial complexities of O(N 3 logNC) and O(NA logNC), respectively. We also discuss practical implementation issues and computational experience to date. Finally, we show how to implementε-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.