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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| Published in: | Networks Vol. 75; no. 3; pp. 235 - 258 |
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| Language: | English |
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John Wiley & Sons, Inc
01.04.2020
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| 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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| AbstractList | 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. 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 ‐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. |
| Author | Rebennack, Steffen Singh, Bismark Prokopyev, Oleg A. |
| Author_xml | – sequence: 1 givenname: Steffen orcidid: 0000-0002-8501-2785 surname: Rebennack fullname: Rebennack, Steffen email: steffen.rebennack@kit.edu organization: Institute for Operations Research, Karlsruhe Institute of Technology – sequence: 2 givenname: Oleg A. orcidid: 0000-0003-2888-8630 surname: Prokopyev fullname: Prokopyev, Oleg A. organization: University of Pittsburgh – sequence: 3 givenname: Bismark surname: Singh fullname: Singh, Bismark organization: Friedrich‐Alexander‐Universität Erlangen‐Nürnberg |
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| Cites_doi | 10.1007/11672142_16 10.1002/net.3230070406 10.1137/1.9780898718799 10.1007/s10107-015-0884-3 10.1145/3274662 10.1007/s10107-005-0638-8 10.1287/moor.1060.0237 10.1137/S0097539792225297 10.1007/11496915_24 10.1515/9781400875184 10.1007/BF01386316 10.1002/net.3230140307 10.1016/j.ejor.2016.12.005 10.1080/03155986.1982.11731876 10.1007/s101070100259 10.1137/S0036144501387141 10.1137/0117061 10.1016/j.cor.2013.01.004 10.1145/234533.234534 10.1007/978-3-642-21527-8_35 10.1007/s10107-013-0742-0 10.1002/rsa.20079 10.1007/s00453-011-9596-0 10.4153/CJM-1956-045-5 10.1287/mnsc.2015.2420 10.1016/j.ejor.2004.10.011 10.1007/BF01585929 10.1137/0601040 |
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| SubjectTerms | Algorithms Benders decomposition combinatorial optimization complexity Decomposition Mathematical analysis Mathematical programming Matrix methods minimum s − t cut problem total unimodularity two‐stage stochastic programming |
| Title | Two‐stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms |
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