Improvements on the Push - Relabel Method : Excess Scaling

We now know about both the methods based on the idea of Ford and Fulkerson (which gave rise to Edmonds and Karp’s algorithm, as well as others) in addition to the PUSH-RELABEL method of Goldberg and Tarjan. However, we see that both of these algorithms have a bottleneck procedure; for Ford and Fulkerson’s algorithm, efficiently finding good augmenting paths is a challenge that many have worked on (for a comprehensive list refer to [2]). For push-relabel algorithms, the bottleneck procedure has always been reducing the amount of nonsaturating pushes executed in the main body of the algorithm. The Relablel-to-Front algorithm reduced the number of nonsaturatng pushes to O(n3) through using a FIFO vertex selection rule [2], while Goldberg and Tarjan improved the running time of their original algorithm to O(mn + log(n2/m)) by using Sleator and Tarjan’s Dynamic Trees data structure to maintain the dynamic transitive closure of the flow network [1]. In this paper, Ahuja and Orlin further improve the running time of the push-relabel method by employing a scaling technique first developed by Edmonds and Karp [3], which introduced the concept of ∆-scaling phases as a method of considering only a certain subset of arcs and/or vertices in the residual flow network Gf . In this synopsis, we will first examine the generic push-relabel algorithm and take note of certain facts surrounding its time complexity. Then, we will consider the original scaling technique demonstrated by Edmonds and Karp in [3], and show how their notion of ∆-scaling phases allows a significant improvement to be made in finding good augmenting paths. Finally, we show how Ahuja and Orlin modified this scaling technique to create an ingenious vertex selection rule for the push-relabel algorithms in [1].