Monotone Boolean dualization is in co-NP[log2n]

Department of Mathematics, University of Patras, Rhion, West Greece, Greece
Information Processing Letters (Impact Factor: 0.55). 01/2003; 85(1):1-6. DOI: 10.1016/S0020-0190(02)00346-0
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In 1996, Fredman and Khachiyan [J. Algorithms 21 (1996) 618–628] presented a remarkable algorithm for the problem of checking the duality of a pair of monotone Boolean expressions in disjunctive normal form. Their algorithm runs in no(logn) time, thus giving evidence that the problem lies in an intermediate class between P and co-NP. In this paper we show that a modified version of their algorithm requires deterministic polynomial time plus O(log2n) nondeterministic guesses, thus placing the problem in the class co-NP[log2n]. Our nondeterministic version has also the advantage of having a simpler analysis than the deterministic one.

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    • "Later, decomposition methods giving rise to trees of polylogarithmic depth were published. In particular, the methods of Kavvadias and Stavropoulos [34] as well as the two methods by Elbassioni in [12] give rise to decomposition trees of polylogarithmic depth. Finally, decomposition methods yielding trees of logarithmic depth were presented by Gaur [17] (see also Gaur and Krishnamurti [18]), and, more recently, by Boros and Makino [4]. "
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    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.
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    • "In pioneering work [22], Fredman and Khachiyan gave an algorithm for solving this problem whose running time is O(n 2 ) + N o(log N) , where N = |F| + |G|, thus providing strong evidence that this decision problem is unlikely to be NP-hard. Eiter et al. [17] [18], and independently Kavvadias and Stavropoulos [36] [35], used the results of [22] to show that the duality of a pair of monotone Boolean functions can be checked in polynomial time with limited nondeterminism, i.e., with a polylogarithmic number of suitably guessed bits, putting thus the problem in the class co-2 P (see e.g., [40] for the definition of this class). Gaur and Krishnamurti [25] gave another algorithm for the equivalent problem of checking if a Boolean function is self-dual (i.e., if f d (x) = f (x)), and showed it to run in time O(N 2 log N+2 ) in general, and in polynomial time on random instances. "
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    • "Hence, Monet and Monet have many applications in such different fields like artificial intelligence and logic [6] [7], computational biology [3], database theory [18], data mining and machine learning [12], mobile communication systems [22], distributed systems [10], and graph theory [13] [16]. The currently best known algorithms for Monet run in quasi-polynomial time or use O(log 2 n) nondeterministic bits [8] [9] [14]. "
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