Compact representations of all members of an independence system

  • Utz-Uwe Haus
  • Carla Michini


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 \(\mathcal {O}(\log n)\) the maximum width of the BDD is bounded by \(\mathcal {O}(n)\). 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.


Compact representations Boolean optimization Enumeration Stable sets Discrete relaxations Decision diagrams 

Mathematics Subject Classifications (2010)

52B99 90C27 05A15 05C30 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Cray EMEA Research LabBaselSwitzerland
  2. 2.WID, OptimizationUniversity of Wisconsin-MadisonMadisonUSA

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