Two-stage stochastic minimum s − t cut problems: formulations, complexity and decomposition algorithms
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 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.
Benders decomposition, combinatorial optimization, complexity, minimum s − t cut problem, total unimodularity, two-stage stochastic programming
235-258
Rebennack, Steffen
d47be0b1-2fbb-4993-955c-d87e00b9266f
Prokopyev, Oleg A.
35f19da2-06b6-4000-96ab-d85ff1c74e07
Singh, Bismark
9d3fc6cb-f55e-4562-9d5f-42f9a3ddd9a1
1 April 2020
Rebennack, Steffen
d47be0b1-2fbb-4993-955c-d87e00b9266f
Prokopyev, Oleg A.
35f19da2-06b6-4000-96ab-d85ff1c74e07
Singh, Bismark
9d3fc6cb-f55e-4562-9d5f-42f9a3ddd9a1
Rebennack, Steffen, Prokopyev, Oleg A. and Singh, Bismark
(2020)
Two-stage stochastic minimum s − t cut problems: formulations, complexity and decomposition algorithms.
Networks, 75 (3), .
(doi:10.1002/net.21922).
Abstract
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.
Text
Networks - 2019 - Rebennack - Two‐stage stochastic minimum s t cut problems Formulations complexity and decomposition
- Version of Record
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Accepted/In Press date: 3 November 2019
e-pub ahead of print date: 29 November 2019
Published date: 1 April 2020
Additional Information:
Funding Information:
information U.S. Air Force Office of Scientific Research, FA9550-08-1-0268 and FA9550-11-1-0037S.R. thanks Klaus Truemper for his discussions on Section 2.2. We thank Ömer Özümerzifon (KIT) for his work on the C++ implementations. We also thank three anonymous reviewers and the associate editor for their detailed and constructive comments. O.A.P. was partially supported by the U.S. Air Force Office of Scientific Research Grants FA9550-08-1-0268 and FA9550-11-1-0037. B.S. thanks Sandia National Laboratories, USA for the part of his time spent there as a postdoctoral appointee. S.R. thanks Klaus Truemper for his discussions on Section 2.2 .
Keywords:
Benders decomposition, combinatorial optimization, complexity, minimum s − t cut problem, total unimodularity, two-stage stochastic programming
Identifiers
Local EPrints ID: 472276
URI: http://eprints.soton.ac.uk/id/eprint/472276
ISSN: 0028-3045
PURE UUID: c62ed51b-0210-41ca-8684-16f57378fb6f
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Date deposited: 30 Nov 2022 17:43
Last modified: 18 Mar 2024 04:08
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Contributors
Author:
Steffen Rebennack
Author:
Oleg A. Prokopyev
Author:
Bismark Singh
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