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Two‐stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms
Author(s) -
Rebennack Steffen,
Prokopyev Oleg A.,
Singh Bismark
Publication year - 2020
Publication title -
networks
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.977
H-Index - 64
eISSN - 1097-0037
pISSN - 0028-3045
DOI - 10.1002/net.21922
Subject(s) - mathematics , benders' decomposition , mathematical optimization , stochastic programming , linear programming , algorithm , decomposition , tree (set theory) , maximum flow problem , computer science , combinatorics , ecology , biology
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.