Generalized roof duality
(2012) In Discrete Applied Mathematics 160(16-17). p.2419-2434- Abstract
- The roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds. We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are... (More)
- The roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds. We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are defined to be the ones that give the maximum lower bound. We show how submodular relaxations can efficiently be constructed in order to compute the generalized roof dual bound for general cubic and quartic pseudo-boolean functions. Further, we prove that important properties such as persistency still hold, which allows us to determine optimal values for some of the variables. From a practical point of view, we experimentally demonstrate that the technique outperforms the state of the art for a wide range of applications, both in terms of lower bounds and in the number of assigned variables. (C) 2012 Elsevier B.V. All rights reserved. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3132755
- author
- Kahl, Fredrik LU and Strandmark, Petter LU
- organization
- publishing date
- 2012
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Roof duality, Higher-order, MRF, Computer vision
- in
- Discrete Applied Mathematics
- volume
- 160
- issue
- 16-17
- pages
- 2419 - 2434
- publisher
- Elsevier
- external identifiers
-
- wos:000308849200009
- scopus:84865103957
- ISSN
- 1872-6771
- DOI
- 10.1016/j.dam.2012.06.009
- language
- English
- LU publication?
- yes
- id
- 56b71a24-0da0-4f44-b3e4-8bc07f2e80a1 (old id 3132755)
- date added to LUP
- 2016-04-01 10:52:15
- date last changed
- 2022-01-26 03:17:49
@article{56b71a24-0da0-4f44-b3e4-8bc07f2e80a1,
abstract = {{The roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds. We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are defined to be the ones that give the maximum lower bound. We show how submodular relaxations can efficiently be constructed in order to compute the generalized roof dual bound for general cubic and quartic pseudo-boolean functions. Further, we prove that important properties such as persistency still hold, which allows us to determine optimal values for some of the variables. From a practical point of view, we experimentally demonstrate that the technique outperforms the state of the art for a wide range of applications, both in terms of lower bounds and in the number of assigned variables. (C) 2012 Elsevier B.V. All rights reserved.}},
author = {{Kahl, Fredrik and Strandmark, Petter}},
issn = {{1872-6771}},
keywords = {{Roof duality; Higher-order; MRF; Computer vision}},
language = {{eng}},
number = {{16-17}},
pages = {{2419--2434}},
publisher = {{Elsevier}},
series = {{Discrete Applied Mathematics}},
title = {{Generalized roof duality}},
url = {{http://dx.doi.org/10.1016/j.dam.2012.06.009}},
doi = {{10.1016/j.dam.2012.06.009}},
volume = {{160}},
year = {{2012}},
}