A Cut-Based Algorithm for the Nonlinear Dual of the Minimum Cost Network Flow
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Authors: | Ravindra K Ahuja Dorit S Hochbaum and James B Orlin |
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Affiliation: | (1) Industrial & Systems Engineering, University of Florida, Gainesville, FL 32611, USA;(2) Industrial Engineering & Operations Research and Walter A. Haas School of Business, University of California, Berkeley, CA 94720, USA;(3) Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139, USA |
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Abstract: | We consider a convex, or nonlinear, separable minimization problem
with constraints that are dual to the minimum cost network flow problem.
We show how to reduce this problem to a polynomial number of
minimum s,t-cut problems. The solution of the reduced problem
utilizes the technique for solving integer programs on monotone
inequalities in three variables, and a so-called
proximity-scaling technique that reduces a convex problem to its
linear objective counterpart. The problem is solved in this case
in a logarithmic number of calls, O(log U), to a minimum cut
procedure, where U is the range of the variables. For a convex
problem on n variables the minimum cut is solved on a graph
with O(n2) nodes. Among the consequences of this result is a
new cut-based scaling algorithm for the minimum cost network flow
problem. When the objective function is an arbitrary nonlinear
function we demonstrate that this constrained problem is solved in
pseudopolynomial time by applying a minimum cut procedure to a
graph on O(nU) nodes. |
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Keywords: | Nonlinear integer programming Convex integer
programming Total unimodularity Minimum cut Network flow |
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