Two‐stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms

We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the to...

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Bibliographic Details
Published in:Networks Vol. 75; no. 3; pp. 235 - 258
Main Authors: Rebennack, Steffen, Prokopyev, Oleg A., Singh, Bismark
Format: Journal Article
Language:English
Published: Hoboken, USA John Wiley & Sons, Inc 01.04.2020
Wiley Subscription Services, Inc
Subjects:
Algorithms
Benders decomposition
combinatorial optimization
complexity
Decomposition
Mathematical analysis
Mathematical programming
Matrix methods
minimum s − t cut problem
total unimodularity
two‐stage stochastic programming
ISSN:0028-3045, 1097-0037
Online Access:Get full text
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Summary:We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP‐hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max‐flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max‐flow structure of the subproblems.
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ISSN:0028-3045
1097-0037
DOI:10.1002/net.21922