Minimum cost multiflows in undirected networks

AV Karzanov - Mathematical Programming, 1994 - Springer
AV Karzanov
Mathematical Programming, 1994Springer
Abstract Let N=(G, T, c, a) be a network, where G is an undirected graph, T is a distinguished
subset of its vertices (called terminals), and each edge e of G has nonnegative integer-
valued capacity c (e) and cost a (e). The minimum cost maximum multi (commodity) flow
problem (*) studied in this paper is to find ac-admissible multiflow f in G such that:(i) f is
allowed to contain partial flows connecting any pairs of terminals,(ii) the total value of f is as
large as possible, and (iii) the total cost of f is as small as possible, subject to (ii). This …
Abstract
LetN = (G, T, c, a) be a network, whereG is an undirected graph,T is a distinguished subset of its vertices (calledterminals), and each edgee ofG has nonnegative integer-valuedcapacity c(e) andcost a(e). Theminimum cost maximum multi(commodity)flow problem (*) studied in this paper is to find ac-admissible multiflowf inG such that: (i)f is allowed to contain partial flows connecting any pairs of terminals, (ii) the total value off is as large as possible, and (iii) the total cost off is as small as possible, subject to (ii). This generalizes, on one hand, the undirected version of the classical minimum cost maximum flow problem (when |T| = 2), and, on the other hand, the problem of finding a maximum fractional packing ofT-paths (whena ≡ 0). Lovász and Cherkassky independently proved that the latter has a half-integral optimal solution.
A pseudo-polynomial algorithm for solving (*) has been developed earlier and, as its consequence, the theorem on the existence of a half-integral optimal solution for (*) was obtained. In the present paper we give a direct, shorter, proof of this theorem. Then we prove the existence of a half-integral optimal solution for the dual problem. Finally, we show that half-integral optimal primal and dual solutions can be designed by a combinatorial strongly polynomial algorithm, provided that some optimal dual solution is known (the latter can be found, in strongly polynomial time, by use of a version of the ellipsoid method).
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