A Strongly Polynomial Cut Canceling Algorithm for Minimum Cost Submodular Flow

This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary min-cost flow. The algorithm scales a relaxed optimality parameter and creates a second, inner relaxation that...

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Published in:SIAM Journal on Discrete Mathematics Vol. 19; no. 2; pp. 304 - 320
Main Authors: Iwata, Satoru, McCormick, S. Thomas, Shigeno, Maiko
Format: Journal Article
Language:English
Japanese
Published: Philadelphia Society for Industrial & Applied Mathematics (SIAM) 01.01.2005
Society for Industrial and Applied Mathematics
Subjects:
Algorithms
Applied mathematics
Costs
Integer programming
Linear algebra
Mathematical programming
Optimization
ISSN:0895-4801, 1095-7146
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Abstract This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary min-cost flow. The algorithm scales a relaxed optimality parameter and creates a second, inner relaxation that is a kind of submodular max flow problem. The outer relaxation uses a novel technique for relaxing the submodular constraints that allows our previous proof techniques to work. The algorithm uses the min cuts from the max flow subproblem as the relaxed most positive cuts it chooses to cancel. We show that this algorithm needs to cancel only ${\mathrm O}(n^3)$ cuts per scaling phase, where $n$ is the number of nodes. Furthermore, we show how to slightly modify this algorithm to get a strongly polynomial running time. Finally, we briefly show how to extend this algorithm to the separable convex cost case and that the same technique can be used to construct a polynomial time maximum mean cut canceling algorithm for submodular flow.
AbstractList This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary min-cost flow. The algorithm scales a relaxed optimality parameter and creates a second, inner relaxation that is a kind of submodular max flow problem. The outer relaxation uses a novel technique for relaxing the submodular constraints that allows our previous proof techniques to work. The algorithm uses the min cuts from the max flow subproblem as the relaxed most positive cuts it chooses to cancel. We show that this algorithm needs to cancel only ${\mathrm O}(n^3)$ cuts per scaling phase, where $n$ is the number of nodes. Furthermore, we show how to slightly modify this algorithm to get a strongly polynomial running time. Finally, we briefly show how to extend this algorithm to the separable convex cost case and that the same technique can be used to construct a polynomial time maximum mean cut canceling algorithm for submodular flow.
Author S. Thomas McCormick
Satoru Iwata
Maiko Shigeno
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  fullname: McCormick, S. Thomas
– sequence: 3
  givenname: Maiko
  surname: Shigeno
  fullname: Shigeno, Maiko
BackLink https://cir.nii.ac.jp/crid/1873116917527209472$$DView record in CiNii
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Snippet This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut...
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SubjectTerms Algorithms
Applied mathematics
Costs
Integer programming
Linear algebra
Mathematical programming
Optimization
Title A Strongly Polynomial Cut Canceling Algorithm for Minimum Cost Submodular Flow
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