Bottleneck flows in unit capacity networks

Abstract

The bottleneck network flow problem (BNFP) is a generalization of several well-studied bottleneck problems such as the bottleneck transportation problem (BTP), bottleneck assignment problem (BAP), bottleneck path problem (BPP), and so on. The BNFP can easily be solved as a sequence of O ( log n ) maximum flow problems on almost unit capacity networks. We observe that this algorithm runs in O ( min { m 3 / 2 , n 2 / 3 m } log n ) time by showing that the maximum flow problem on an almost unit capacity graph can be solved in O ( min { m 3 / 2 , n 2 / 3 m } ) time. We then propose a faster algorithm to solve the unit capacity BNFP in O ( min { m ( n log n ) 2 / 3 , m 3 / 2 log n } ) time, an improvement by a factor of at least log n 3 . For dense graphs, the improvement is by a factor of log n . On unit capacity simple graphs, we show that BNFP can be solved in O ( m n log n ) time, an improvement by a factor of log n . As a consequence we have an O ( m n log n ) algorithm for the BTP with unit arc capacities.