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.
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More information
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.
Funding Information:
S.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.
Publisher Copyright:
© 2019 The Authors. Networks published by Wiley Periodicals, Inc.
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: 23 Feb 2023 03:29
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Contributors
Author:
Steffen Rebennack
Author:
Oleg A. Prokopyev
Author:
Bismark Singh
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