Monotone Boolean dualization is in co-NP[log2n]

Kavvadias, Dimitris J.; Stavropoulos, Elias C.

doi:10.1016/s0020-0190(02)00346-0

Cited by 28 publications

(21 citation statements)

References 9 publications

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“…H, i.e., a transversal of G that is also an independent set of H. In fact, many algorithms in the literature follow this approach (see, e.g., [4,16,19,22,24,40]). These algorithms try to build such a new transversal by successively including vertices in and excluding vertices from a candidate for a new transversal.…”

Section: Decomposition Principlesmentioning

confidence: 99%

“…Among the vast literature published so far on the topic, we find that computing the formula CNF (f d ) from CNF (f ) (or its decision flavour, that is, given two CNF formulas deciding whether they are dual) was studied, for example, by Eiter et al [16,17]; and that computing the formula DNF (f d ) from DNF (f ) (or its decision flavour, that is, given two DNF formulas deciding whether they are dual) was studied, for example, by Fredman and Khachiyan [22], and Kavvadias and Stavropoulos [39,40].…”

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confidence: 99%

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Achieving new upper bounds for the hypergraph duality problem through logic

Gottlob

Malizia

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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The hypergraph duality problem Dual is defined as follows: given two simple hypergraphs G and H, decide whether H consists precisely of all minimal transversals of G (in which case we say that G is the dual of H, or, equivalently, the transversal hypergraph of H). This problem is equivalent to deciding whether two given non-redundant monotone DNFs/CNFs are dual. It is known that Dual, the complementary problem to Dual, is in GC(log 2 n, PTIME), where GC(f (n), C) denotes the complexity class of all problems that after a nondeterministic guess of O(f (n)) bits can be decided (checked) within complexity class C. It was conjectured that Dual is in GC(log 2 n, LOGSPACE). In this paper we prove this conjecture and actually place the Dual problem into the complexity class GC(log 2 n, TC 0 ) which is a subclass of GC(log 2 n, LOGSPACE). We here refer to the logtime-uniform version of TC 0 , which corresponds to FO(COUNT), i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for Dual that requires to guess O(log 2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT). From this result, by the well known inclusion TC 0 ⊆ LOGSPACE, it follows that Dual belongs also to DSPACE[log 2 n]. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs G and H, computes in quadratic logspace a transversal of G missing in H.

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Section: Decomposition Principlesmentioning

confidence: 99%

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confidence: 99%

Achieving new upper bounds for the hypergraph duality problem through logic

Gottlob

Malizia

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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“…Computational complexity of this problem has now been extensively studied [7,8,13,10,27,26,11,12], and many important problems from various fields of computer science have been shown to be computationally equivalent to this problem. Some of these problems are: from relational databases the problem fd-relation equivalence, which is checking whether a given set of functional dependencies that is in Boyce-Codd Normal Form is a cover of a given relation instance [8], the problem additional key for relation instances, which is the problem of checking whether an additional key exists for a given relation instance and a set of minimal keys thereof [8], and from logic the problem monotone Boolean duality, which is checking whether two monotone Boolean functions given by their irredundant disjunctive normal forms are mutually dual [13].…”

Section: Definition 12mentioning

confidence: 99%

On the complexity of enumerating pseudo-intents

Distel

Sertkaya²

2011

Discrete Applied Mathematics

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a b s t r a c tWe investigate whether the pseudo-intents of a given formal context can efficiently be enumerated. We show that they cannot be enumerated in a specified lexicographic order with polynomial delay unless P = NP. Furthermore we show that if the restriction on the order of enumeration is removed, then the problem becomes at least as hard as enumerating minimal transversals of a given hypergraph. We introduce the notion of minimal pseudo-intents and show that recognizing minimal pseudo-intents is polynomial. Despite their less complicated nature, surprisingly it turns out that minimal pseudo-intents cannot be enumerated in output-polynomial time unless P = NP.

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“…See [Hag08,Chapter 3] for a more detailed list of equivalent problems and possible applications. The currently best known Monet algorithms run in quasi-polynomial n o(log n) time or use O(log 2 n) nondeterministic bits [EGM03,FK96,KS03]. Thus, on the one hand, Monet is probably not coNPcomplete, but on the other hand a polynomial time algorithm is not yet known.…”

Section: Introductionmentioning

confidence: 99%

Experimental comparison of the two Fredman-Khachiyan-algorithms

Hagen¹,

Horatschek²,

Mundhenk³

2009

2009 Proceedings of the Eleventh Workshop on Algorithm Engineering and Experiments (ALENEX)

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We experimentally compare the two algorithms A and B by Fredman and Khachiyan [FK96] for the problem Monet-given two monotone Boolean formulas ϕ in DNF and ψ in CNF, decide whether they are equivalent. Currently, algorithm B is the Monet algorithm with the best known worst-case performance. However, there is no experimental evaluation of its practical performance yet, mainly due to the following two reasons. Firstly, implementation of algorithm B is usually considered to be more involved than for algorithm A. Secondly and probably more importantly, there is the assumption that the operations performed by algorithm B to ensure recursion on smaller sub-problems do only pay off theoretically.In this paper, we contrast this assumption by experimentally showing algorithm B to be competitive and even superior to algorithm A on many instances.

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Monotone Boolean dualization is in co-NP[log2n]

Cited by 28 publications

References 9 publications

Achieving new upper bounds for the hypergraph duality problem through logic

Achieving new upper bounds for the hypergraph duality problem through logic

On the complexity of enumerating pseudo-intents

Experimental comparison of the two Fredman-Khachiyan-algorithms

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