New Results on Monotone Dualization and Generating Hypergraph Transversals

Eiter, Thomas; Gottlob, Georg; Makino, Kazuhisa

doi:10.1137/s009753970240639x

Cited by 106 publications

(46 citation statements)

References 48 publications

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“…The dualization of a monotone Boolean function is ubiquitous in many areas of computer science including database theory, logic and artificial intelligence [EG95, GMKT97,EGM03,NP12]. When defined on Boolean lattices, the problem is equivalent to the enumeration of the minimal transversals of a hypergraph, arguably one of the most important open problems in algorithmic enumeration by now [EG95,EMG08].…”

Section: Introductionmentioning

confidence: 99%

Dualization in Lattices Given by Implicational Bases

Defrain

Nourine

2019

Formal Concept Analysis

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It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP. In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. Then we show using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent conclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one.

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Section: Introductionmentioning

confidence: 99%

Dualization in Lattices Given by Implicational Bases

Defrain

Nourine

2019

Formal Concept Analysis

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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%

“…Then, the procedure checks again whether σ is a covering assignment (line 14). Otherwise, the procedure tests whether σ is a witness (line 16). If this is not the case, then the procedure tries to include each of the free vertices of σ as a critical vertex with an edge from Sep(σ) witnessing its criticality (lines [17][18][19].…”

Section: Hence If T Is a New Transversal Of G Wrt H Then For Anmentioning

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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“…By Lemma 5, the total time for this generation is polynomial in |V |, |E|, and |M 2 (G)|. Since the problem is self-reducible, we can convert this output polynomial generation algorithm to an incremental one, see [3,8,18] for more details.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

2006

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A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, we obtain a polynomial delay algorithm for listing the anti-vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems,

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New Results on Monotone Dualization and Generating Hypergraph Transversals

Cited by 106 publications

References 48 publications

Dualization in Lattices Given by Implicational Bases

Dualization in Lattices Given by Implicational Bases

Achieving new upper bounds for the hypergraph duality problem through logic

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

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