Dual coordinate step methods for linear network flow problems (Q1115790)

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 \(\epsilon\)-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 \(\epsilon\)-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 \(\epsilon\)- 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.