On the Complexity of Some Ordering Problems

Bollig, Beate

doi:10.1007/978-3-662-44465-8_11

Cited by 6 publications

(8 citation statements)

References 26 publications

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“…For this we present another variable ordering π ′ and we show that π ′ -OBDD(f n ) is smaller than π -OBDD(f n ). The variable orderings π = (x (3) n−1 ,…”

Section: Definition 26mentioning

confidence: 99%

“…It is not difficult to see that besides the choice of the variable ordering, the encoding of the vertices can influence the size of the OBDD-based graph representations. (For a simple example see, e.g., Section 4 in [3].) Up to now to the best of our knowledge there is no efficient algorithm known how to encode the vertices space efficiently with respect to the OBDD representation size of the input graph.…”

Section: Introductionmentioning

confidence: 99%

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On the OBDD representation of some graph classes

Bollig

Bury

2016

Discrete Applied Mathematics

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a b s t r a c tOrdered Binary Decision Diagrams (OBDDs) are a popular data structure for Boolean functions. Given the rapid growth of application-based networks, an heuristic approach to deal with very large structured graphs are implicit OBDD-based graph algorithms. Vertices of an input graph are binary encoded and the edge set is represented by its characteristic function. Since OBDDs are able to take advantage of the presence of regular substructures, this approach leads sometimes to sublinear graph representations. By simple counting arguments it is easy to see that almost all graphs on N vertices cannot be represented by OBDDs of polylogarithmic size with respect to N. On the other hand, very simply structured graphs like grid graphs have small OBDD representations. Here, the investigation for which significant graph classes succinct OBDD representations are possible is continued. Using a new method how to use the structure of the adjacency matrix representation of a graph to count subfunctions of the characteristic Boolean function of its edge set, known upper bounds on the OBDD size of interval graphs are improved. Furthermore, the OBDD size for interval bigraphs, (bi)convex graphs, bipartite permutation graphs, chain graphs, threshold graphs, and trees is analyzed. Except for interval graphs, interval bigraphs, and convex graphs the presented bounds are tight. Finally, regular graphs are considered and examples are presented with small and large OBDD size. Moreover, a variable ordering often used in the implicit setting is investigated.

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“…For this we present another variable ordering π ′ and we show that π ′ -OBDD(f n ) is smaller than π -OBDD(f n ). The variable orderings π = (x (3) n−1 ,…”

Section: Definition 26mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the OBDD representation of some graph classes

Bollig

Bury

2016

Discrete Applied Mathematics

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“…It is easy to see that the size of a quasi-reduced OBDD representing f k with respect to a variable ordering (1) , v π (2) , . .…”

Section: The Polynomial Time Reductionsmentioning

confidence: 99%

“…Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.Preliminary versions of parts of this paper have been appeared in [2,3] The author is supported by DFG project BO 2755/1-2.Beate Bollig …”

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confidence: 99%

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On the Minimization of (Complete) Ordered Binary Decision Diagrams

Bollig

2015

Theory Comput Syst

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Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.Preliminary versions of parts of this paper have been appeared in [2,3] The author is supported by DFG project BO 2755/1-2.Beate Bollig

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