A polynomial algorithm for abstract maximum flow
- Univ. of British Columbia, Vancouver, British Columbia (Canada)
Ford and Fulkerson`s original 1956 max flow/min cut paper formulated max flow in terms of flows on paths, rather than the more familiar flows on arcs. In 1974 Hoffman pointed out that Ford and Fulkerson`s original proof was quite abstract, and applied to a wide range of max flow-like problems. In this abstract model we have capacitated elements, and linearly ordered subsets of elements called paths. When two paths share an element ({open_quote}cross{close_quote}), then there must be a path that is a subset of the first path up to the cross, and a subset of the second path after the cross. (Hoffman`s generalization of) Ford and Fulkerson`s proof showed that the max flow/min cut theorem still holds under this weak assumption. However, this proof is non-constructive. To get an algorithm, we assume that we have an oracle whose input is an arbitrary subset of elements, and whose output is either a path contained in that subset, or the statement that no such path exists. We then use complementary slackness to show how to augment any feasible set of path flows to a set with a strictly larger total flow value using a polynomial number of calls to the oracle. Then standard scaling techniques yield an overall polynomial algorithm for finding both a max flow and a min cut. Hoffman`s paper actually considers a sort of supermodular objective on the path flows, which allows him to include transportation problems and thus rain-cost flow in his frame-work. We also discuss extending our algorithm to this more general case.
- OSTI ID:
- 416834
- Report Number(s):
- CONF-960121-; TRN: 96:005887-0057
- Resource Relation:
- Conference: 7. annual ACM-SIAM symposium on discrete algorithms, Atlanta, GA (United States), 28-30 Jan 1996; Other Information: PBD: 1996; Related Information: Is Part Of Proceedings of the seventh annual ACM-SIAM symposium on discrete algorithms; PB: 596 p.
- Country of Publication:
- United States
- Language:
- English
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