Article

# Monotone Boolean dualization is in co-NP[log2n]

University of Patras, Department of Mathematics, GR-265 00 Patras, Greece; University of Patras, Computer Engineering & Informatics Department, GR-265 00 Patras, Greece

Information Processing Letters (Impact Factor: 0.49). 01/2003; DOI: 10.1016/S0020-0190(02)00346-0 Source: CiteSeer

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**ABSTRACT:**We consider the computation of all respectively a polynomial subset of the explanations of an abductive query from a Horn theory, and pay particular attention to whether the query is a positive or negative letter, the explanation is based on literals from an assumption set, and the Horn theory is represented in terms of formulas or characteristic models. We derive tractability results, one of which refutes a conjecture by Selman and Levesque, as well as intractability results, and furthermore also semi-tractability results in terms of solvability in quasi-polynomial time. Our results complement previous results in the literature, and elucidate the computational complexity of generating the set of explanations.J. ACM. 01/2007; 54. - [Show abstract] [Hide abstract]

**ABSTRACT:**The monotone duality problem is defined as follows: Given two monotone formulas f and g in iredundant DNF, decide whether f and g are dual. This problem is the same as duality testing for hypergraphs, that is, checking whether a hypergraph H consists of precisely all minimal transversals of a simple hypergraph G. By exploiting a recent problem-decomposition method by Boros and Makino (ICALP 2009), we show that duality testing for hypergraphs, and thus for monotone DNFs, is feasible in DSPACE[log^2 n], i.e., in quadratic logspace. As the monotone duality problem is equivalent to a number of problems in the areas of databases, data mining, and knowledge discovery, the results presented here yield new complexity results for those problems, too. For example, it follows from our results that whenever for a Boolean-valued relation (whose attributes represent items), a number of maximal frequent itemsets and a number of minimal infrequent itemsets are known, then it can be decided in quadratic logspace whether there exist additional frequent or infrequent itemsets.12/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we present fixed-parameter algorithms for the problem Dual—given two hypergraphs, decide if one is the transversal hypergraph of the other—and related problems. In the first part, we consider the number of edges of the hypergraphs, the maximum degree of a vertex, and a vertex complementary degree as our parameters. In the second part, we use an Apriori approach to obtain FPT results for generating all maximal independent sets of a hypergraph, all mini- mal transversals of a hypergraph, and all maximal frequent sets where parameters bound the intersections or unions of edges.Grohe, Martin; Niedermeier, Rolf: Parameterized and Exact Computation : Third International Workshop, IWPEC 2008, Springer, 91-102 (2008). 01/2008;

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