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Compact representations of all members of an independence system

Compact representations of all members of an independence system It is well known that (reduced, ordered) binary decision diagrams (BDDs) can sometimes be compact representations of the full solution set of Boolean optimization problems. Recently they have been suggested to be useful as discrete relaxations in integer and constraint programming (Hoda et al. 2010). We show that for every independence system there exists a top-down (i.e., single-pass) construction rule for the BDD. Furthermore, for packing and covering problems on n variables whose bandwidth is bounded by O ( log n ) $\mathcal {O}(\log n)$ the maximum width of the BDD is bounded by O ( n ) $\mathcal {O}(n)$ . We also characterize minimal widths of BDDs representing the set of all solutions to a stable set problem for various basic classes of graphs. Besides implicitly enumerating or counting all solutions and optimizing a class of nonlinear objective functions that includes separable functions, the results can be applied for effective evaluation of generating functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Mathematics and Artificial Intelligence Springer Journals

Compact representations of all members of an independence system

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References (28)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing Switzerland
Subject
Computer Science; Artificial Intelligence (incl. Robotics); Mathematics, general; Computer Science, general; Complex Systems
ISSN
1012-2443
eISSN
1573-7470
DOI
10.1007/s10472-016-9496-8
Publisher site
See Article on Publisher Site

Abstract

It is well known that (reduced, ordered) binary decision diagrams (BDDs) can sometimes be compact representations of the full solution set of Boolean optimization problems. Recently they have been suggested to be useful as discrete relaxations in integer and constraint programming (Hoda et al. 2010). We show that for every independence system there exists a top-down (i.e., single-pass) construction rule for the BDD. Furthermore, for packing and covering problems on n variables whose bandwidth is bounded by O ( log n ) $\mathcal {O}(\log n)$ the maximum width of the BDD is bounded by O ( n ) $\mathcal {O}(n)$ . We also characterize minimal widths of BDDs representing the set of all solutions to a stable set problem for various basic classes of graphs. Besides implicitly enumerating or counting all solutions and optimizing a class of nonlinear objective functions that includes separable functions, the results can be applied for effective evaluation of generating functions.

Journal

Annals of Mathematics and Artificial IntelligenceSpringer Journals

Published: Mar 11, 2016

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