Finite Termination of "Augmenting Path" Algorithms in the Presence of Irrational Problem Data

This paper considers two similar graph algorithms that work by repeatedly increasing "flow" along "augmenting paths": the Ford-Fulkerson algorithm for the maximum flow problem and the Gale-Shapley algorithm for the stable allocation problem (a many-to-many generalization of the stable matching problem). Both algorithms clearly terminate when given integral input data. For real-valued input data, it was previously known that the Ford-Fulkerson algorithm runs in polynomial time if augmenting paths are chosen via breadth-first search, but that the algorithm might fail to terminate if augmenting paths are chosen in an arbitrary fashion. However, the performance of the Gale-Shapley algorithm on real-valued data was unresolved. Our main result shows that, in contrast to the Ford-Fulkerson algorithm, the Gale-Shapley algorithm always terminates in finite time on real-valued data. Although the Gale-Shapley algorithm may take exponential time in the worst case, it is a popu...

• Type: Conference
• Year: 2006
• Where: ESA
• Authors: Brian C. Dean, Michel X. Goemans, Nicole Immorlica
• Keywords: Algorithm, Algorithms, ESA 2006, Ford-Fulkerson Algorithm, Gale-Shapley Algorithm