Computational aspects of monotone dualization: A brief survey

Abstract

Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence, and Game Theory to mention some of them. It is also one of the few problems whose precise tractability status (in terms of polynomial-time solvability) is still unknown, and now open for more than 25 years. In this paper, we briefly survey computational results for this problem, where we focus on the famous paper by Fredman and Khachiyan (J. Algorithms, 21:618-628, 1996), which showed that the problem is solvable in quasi-polynomial time (and thus most likely not co-NP-hard), as well as on follow-up works. We consider computational aspects including limited nondeterminism, probabilistic computation, parallel and learning-based algorithms, and implementations and experimental results from the literature. The paper closes with open issues for further research.

Dualization in lattices given by ordered sets of irreducibles

“…In [5,7] the author give quasi-polynomial time algorithms for the following cases: each P i is a join semi-lattice of bounded width (any antichain has constant size), each P i is a forest poset in which either the in-degree or the out-degree of each element is constant (see also [6]), each P i is the lattice of intervals defined by a set of intervals on the real line R. In [5,7] a more general dualization problem was stated where each P i is a lattice (with no bounds on its width), the existence of quasipolynomial time algorithms for this case is still an open question. In this paper we prove an upper bound complexity of the latter problem via another long-standing open complexity problem, the minimum implication base (see [2], equivalently SID problem from [21,4]). The most common technique leading to quasi-polynomial time algorithm for duality problems are based on the idea of high frequency based decomposition, first introduced in [10].…”

“…Therefore, the dualization on lattices given by implication bases for distributive lattices is polynomially equivalent to the dualization on lattices given by contexts (ordered sets of irreducible elements), which we study in the next section. The study of dualization problems for lattices given by implication bases is motivated by simple linear-time reciprocal translations of implications to functional dependencies [18] and propositional Horn theories [4].…”

“…Dualization is the transformation of the set of minimal 1 values of a Boolean function to the set of its maximal 0 values or vice versa. Since dualization is equivalent to many important problems in computer and data sciences [4,19,5], the paper [10] on quasi-polynomial dualization algorithm for Boolean lattices was an important breakthrough. It paved the way to generalizations to various classes of structures where dualization in output subexponential time is possible, among them dualization on lattices given by ordered sets of their elements or by products of bounded width lattices, like chains [4,5].…”

Output-Sensitive Algorithms for Enumerating Minimal Transversals for Some Geometric Hypergraphs

“…The enumeration of minimal geometric hitting sets, as the ones described above, arises in various areas such as computational geometry, machine learning, and data mining [14]. Moreover, our efficient enumeration algorithm might be useful, for example, in developing exact algorithms, fixed-parameter tractable algorithms, and polynomial-time approximation schemes for the corresponding optimization problems (see, e.g., [22]).…”

On Berge Multiplication for Monotone Boolean Dualization

“…This is witnessed by the fact that these problems are cited in a rapidly growing body of literature and have been referenced in various survey papers and complexity theory retrospectives, e.g. [12,14,21,29,30,32].…”