Balanced network flows. III. Strongly polynomial augmentation algorithms

Christian Fremuth-Paeger, Dieter Jungnickel
1999 Networks  
We discuss efficient augmentation algorithms for the maximum balanced flow problem which run in O(nm 2 ) time. More explicitly, we discuss a balanced network search procedure which finds valid augmenting paths of minimum length in linear time. The algorithms are based on the famous cardinality matching algorithm given by Micali and Vazirani. A comprehensive description of the double depth first search is included. PRELIMINARIES pairs with residual capacity one. Furthermore, augmentation is
more » ... s done pairwise on complementary paths. The most important part of the augmentation algorithm Balanced flow networks were studied extensively in [4]. is the balanced network search procedure BNS which Solving the maximum balanced flow problem means solvchecks whether a valid augmenting path exists or not. ing the wide range of nonweighted matching problems. The implementations in [5] are highly efficient if the In [5], we gave a large amount of pseudocode for solving maximum flow value is small compared with the size of this problem. In particular, we proposed an augmentation the original graph G, in particular, if G is simple. Howprocedure and a disjoint set union mechanism which will ever, this was not a polynomial time procedure in the be reused here. general setting. Balanced networks are defined on skew-symmetric Here, we show how to derive augmenting paths of graphs. The inherent symmetry partitions the node set minimum length. Our BNS procedure heavily depends into pairs of complementary nodes and the arc set into on the double depth first search method, which was pairs of complementary arcs. The paths used for augmendiscussed by Vazirani [10] before. The idea applies to tation are valid, that is, they avoid complementary arc other BNS procedures as well. For this reason, we describe the general setting. An arc a of a balanced network N which can be acaugsburg.de cessed from the source by a valid path is called strictly The results of this paper form part of the first author's doctoral thesis accessable. If the complementary arc a is also strictly which has been written under the supervision of the second author.
doi:10.1002/(sici)1097-0037(199901)33:1<43::aid-net3>3.0.co;2-6 fatcat:3kuy6jf2zjcn7eehbvfykisdni