Size of ordered binary decision diagrams representing threshold functions

Hosaka, K.; Takenaga, Yasuhiko; Kaneda, Takayuki; Yajima, Shuzo

doi:10.1016/s0304-3975(97)83807-8

Cited by 30 publications

(25 citation statements)

References 7 publications

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Section: Algorithm 4 Generatemwffromswfmentioning

confidence: 83%

“…There exist simple games Γ for which |BDD(Γ )| grows exponentially in terms of n, independently of the order of the variables. For instance, consider the Theorem 4 of Hosaka et al (1997), which shows a subclass of simple games called weighted games, for which |BDD(Γ )| ∈ Ω(2 √ n/2 ). Additionally, there exists a known result by Wegener which says that almost all QOBDDs for general Boolean functions, not only the monotone ones, have size 2 n /(2n) (Wegener, 1994).…”

Section: Representation Sizes and The Conversion Problemmentioning

confidence: 99%

“…There are subclasses of weighted games whose BDDF grows exponentially in terms of n (Hosaka et al, 1997). This behavior is independently of the ordering of the players and the weights in a WRF.…”

Section: Representation Sizes and The Conversion Problemmentioning

confidence: 99%

See 2 more Smart Citations

Forms of representation for simple games: Sizes, conversions and equivalences

Molinero

Riquelme

Serna

2015

Mathematical Social Sciences

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h i g h l i g h t s• We survey the main forms of representation for simple games, regular games and weighted games in literature.• We study the complexity of the conversion problem, i.e., the problem of computing a representation of a simple game given another representation.• We prove that several changes of representation that require exponential time can be solved with polynomial-delay. a b s t r a c tSimple games are cooperative games in which the benefit that a coalition may have is always binary, i.e., a coalition may either win or loose. This paper surveys different forms of representation of simple games, and those for some of their subfamilies like regular games and weighted games. We analyze the forms of representations that have been proposed in the literature based on different data structures for sets of sets. We provide bounds on the computational resources needed to transform a game from one form of representation to another one. This includes the study of the problem of enumerating the fundamental families of coalitions of a simple game. In particular we prove that several changes of representation that require exponential time can be solved with polynomial-delay and highlight some open problems.

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Section: Algorithm 4 Generatemwffromswfmentioning

confidence: 83%

Section: Representation Sizes and The Conversion Problemmentioning

confidence: 99%

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Forms of representation for simple games: Sizes, conversions and equivalences

Molinero

Riquelme

Serna

2015

Mathematical Social Sciences

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“…In [7] the first exponential lower bound of Ω(n2 n 1/2 /2 ) on the OBDD size of a threshold function is proved. Later on this lower bound has been improved up to Ω(n2…”

Section: Results and Related Workmentioning

confidence: 99%

On the size of (generalized) OBDDs for threshold functions

Bollig

2009

Information Processing Letters

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“…• Subsection 3.2: We reproduce the family of PB constraints proposed by Hosaka et al (1994), for which no polynomial-size ROBDD exist. For self-containedness, we give a clearer alternative proof than in the original paper.…”

Section: Introductionmentioning

confidence: 99%

A New Look at BDDs for Pseudo-Boolean Constraints

Abío¹,

Nieuwenhuis²,

Oliveras³

et al. 2012

jair

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Pseudo-Boolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision Diagram (BDD) for the constraint, and then encode the BDD into a propositional formula. These BDD-based approaches have some important advantages, such as not being dependent on the size of the coefficients, or being able to share the same BDD for representing many constraints. We first focus on the size of the resulting BDDs, which was considered to be an open problem in our research community. We report on previous work where it was proved that there are Pseudo-Boolean constraints for which no polynomial BDD exists. We also give an alternative and simpler proof assuming that NP is different from Co-NP. More interestingly, here we also show how to overcome the possible exponential blowup of BDDs by coefficient decomposition. This allows us to give the first polynomial generalized arc-consistent ROBDD-based encoding for Pseudo-Boolean constraints. Finally, we focus on practical issues: we show how to efficiently construct such ROBDDs, how to encode them into SAT with only 2 clauses per node, and present experimental results that confirm that our approach is competitive with other encodings and state-of-the-art Pseudo-Boolean solvers.

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Size of ordered binary decision diagrams representing threshold functions

Cited by 30 publications

References 7 publications

Forms of representation for simple games: Sizes, conversions and equivalences

Forms of representation for simple games: Sizes, conversions and equivalences

On the size of (generalized) OBDDs for threshold functions

A New Look at BDDs for Pseudo-Boolean Constraints

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