Some algorithmic and complexity results for monotone Boolean duality (hypergraph transversal).
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# Some algorithmic and complexity results for monotone Boolean duality (hypergraph transversal). by Philipp Hertel

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Published .
Written in English

## Book details:

Let y be a monotone CNF formula and let 4 be a monotone DNF formula. The problem of determining in polynomial time in the size of y and 4 whether y↾a=4↾ a for all assignments alpha (known as DUAL) has been a longstanding open problem.We show that two popular families of algorithms for DUAL (Berge"s Sequential Method and Fredman and Khachiyan"s Algorithm A ) are really restrictions of the DPLL class of algorithms which has long been studied in relation to SAT. We also present super-polynomial lower bounds for DPLL on two classes of read-once formulae. Finally, we present DPLLCache, DPLL with memorization, and show that DPLLCache is strictly stronger than DPLL and therefore also the Sequential Method and Algorithm A. In fact, we show that DPLLCache can solve all read-once formulae using polynomial size decision trees.

The Physical Object
Pagination61 leaves.
Number of Pages61
ID Numbers
Open LibraryOL19747412M
ISBN 100612914437

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Given the irredundant CNF representation ϕ of a monotone Boolean function f:{0,1} n {0,1}, the dualization problem calls for finding the corresponding unique irredundant DNF representation ψ of (generalized) multiplication method works by repeatedly dividing the clauses of ϕ into (not necessarily disjoint) groups, multiplying-out the clauses in each group, and then reducing the. As the monotone duality problem is equivalent to a number of prob-lems 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).   Abstract: One matrix structure in the area of monotone Boolean functions is defined here. Some of its combinatorial, algebraic and algorithmic properties are derived. On the base of these properties, three algorithms are : Valentin Bakoev. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log 2 n /\log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

Focusing on monotone Boolean functions, we exhibit the existence of near-optimal -approximately (e p from some distribution Dand fis an arbitrary Boolean function. The goal of the agnostic learning algorithm OPT =2 (in other words when noise rate is =2). For simplicity the complexity of an SQ algorithm. oring and monotone self-duality, and results of Linial and Tarsi (Linial and Tarsi ) and Seymour (Seymour ) to study the complexity of resolution like proofs for prov-ing self-duality. First we begin with some deﬁnitions. An hypergraph (V;E) is a collection E of subsets of V. Ele-ments of V are referred to as the vertices and the elements. BOOLEAN FUNCTIONS Theory, Algorithms, and Applications Yves CRAMA and Peter L. HAMMER with contributions by Claude Benzaken, Endre Boros, Nadia Brauner, Martin C. Golumbic, Vladimir Gurvich. Part I. Foundations: 1. Fundamental concepts and applications 2. Boolean equations 3. Prime implicants and minimal DNFs Peter L. Hammer and Alexander Kogan 4. Duality theory Yves Crama and Kazuhisa Makino Part II. Special Classes: 5. Quadratic functions Bruno Simeone 6. Horn functions Endre Boros 7. Orthogonal forms and shellability 8. Regular functions 9. Threshold functions Read-once.

It is not known whether self-duality of monotone boolean functions can be tested in polynomial time, though a quasi-polynomial time algorithm exists. We describe another quasi-polynomial time algorithm for solving the self-duality problem of monotone boolean functions and analyze its average-case behaviour on a set of randomly generated instances. algorithms for the problem of self-duaiity of monotone boolean functions and some related problems. We show that several special cases of the problem can be solved in polynomial tinic. The gcnerai problcrn is showed to be solvabIc in quasi-polynomial time. The problem of determining if a given monotone boolean function is seif-dual arises. We show that the duality of a pair of monotone Boolean functions in disjunctive normal forms can be tested in polylogarithmic time using a quasi-polynomial number of processors. Our decomposition technique yields stronger bounds on the complexity of the problem than those currently known and also allows for generating all minimal transversals of a given hypergraph using only polynomial space. In particular, it is possible to define a partial order set over the set of all monotone Boolean functions which can be represented by an Hasse diagram (Crama and Hammer, ; Cury et al.,