Bottleneck flows in unit capacity networks

Authors:
Abraham P. Punnen;Ruonan Zhang
Affiliations:
Department of Mathematics, Simon Fraser University Surrey, Central City, 250-13450 102nd AV, Surrey, British Columbia, V3T 0A3, Canada;Department of Mathematics, Simon Fraser University Surrey, Central City, 250-13450 102nd AV, Surrey, British Columbia, V3T 0A3, Canada
Venue:
Information Processing Letters
Year:
2009

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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(logn) maximum flow problems on almost unit capacity networks. We observe that this algorithm runs in O(min{m^3^/^2,n^2^/^3m}logn) 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^/^3m}) time. We then propose a faster algorithm to solve the unit capacity BNFP in O(min{m(nlogn)^2^/^3,m^3^/^2logn}) time, an improvement by a factor of at least logn3. For dense graphs, the improvement is by a factor of logn. On unit capacity simple graphs, we show that BNFP can be solved in O(mnlogn) time, an improvement by a factor of logn. As a consequence we have an O(mnlogn) algorithm for the BTP with unit arc capacities.