On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions

AbstractLet f:{0,1}n→{0,1} be a monotone Boolean function whose value at any point x∈{0,1}n can be determined in time t. Denote by c=⋀I∈C⋁i∈Ixi the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d=⋁J∈D⋀j∈Jxj be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets C′⊆C and D′⊆D such that (C′,D′)≠(C,D), a new term in (C⧹C′)∪(D⧹D′) can be found in time O(n(t+n))+mo(logm), where m=|C′|+|D′|. In particular, if f(x) can be evaluated for every x∈{0,1}n in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of ∧,∨-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2polylog(·) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions D′=D or C′=C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of ∧,∨-formulae of depth 3

Incremental complexity of a bi-objective hypergraph transversal problem

The hypergraph transversal problem has been intensively studied, from both a theoretical and a practical point of view. In particular , its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs (A, B), and the aim is to find minimal sets which hit all the hyperedges of A while intersecting a minimal set of hyperedges of B. In this paper, we formalize this problem, link it to a problem on monotone boolean $\land$ -- $\lor$ formulae of depth 3 and study its incremental complexity

On dualization in products of forests, in

Abstract. Let P = P1 ×...×Pn be the product of n partially ordered sets, each with an acyclic precedence graph in which either the in-degree or the out-degree of each element is bounded. Given a subset A⊆P,it is shown that the set of maximal independent elements of A in P can be incrementally generated in quasi-polynomial time. We discuss some applications in data mining related to this dualization problem

Computing knock out strategies in metabolic networks

Given a metabolic network in terms of its metabolites and reactions, our goal is to efficiently compute the minimal knock out sets of reactions required to block a given behaviour. We describe an algorithm which improves the computation of these knock out sets when the elementary modes (minimal functional subsystems) of the network are given. We also describe an algorithm which computes both the knock out sets and the elementary modes containing the blocked reactions directly from the description of the network and whose worst-case computational complexity is better than the algorithms currently in use for these problems. Computational results are included.Comment: 12 page

An Algorithm for Dualization in Products of Lattices and Its Applications

Let \cL=\cL_1×⋅s×\cL_n be the product of n lattices, each of which has a bounded width. Given a subset \cA\subseteq\cL, we show that the problem of extending a given partial list of maximal independent elements of \cA in \cL can be solved in quasi-polynomial time. This result implies, in particular, that the problem of generating all minimal infrequent elements for a database with semi-lattice attributes, and the problem of generating all maximal boxes that contain at most a specified number of points from a given n-dimensional point set, can both be solved in incremental quasi-polynomial time

Enumeration of minimal stoichiometric precursor sets in metabolic networks

Background: What an organism needs at least from its environment to produce a set of metabolites, e.g. target(s) of interest and/or biomass, has been called a minimal precursor set. Early approaches to enumerate all minimal precursor sets took into account only the topology of the metabolic network (topological precursor sets). Due to cycles and the stoichiometric values of the reactions, it is often not possible to produce the target(s) from a topological precursor set in the sense that there is no feasible flux. Although considering the stoichiometry makes the problem harder, it enables to obtain biologically reasonable precursor sets that we call stoichiometric. Recently a method to enumerate all minimal stoichiometric precursor sets was proposed in the literature. The relationship between topological and stoichiometric precursor sets had however not yet been studied. Results: Such relationship between topological and stoichiometric precursor sets is highlighted. We also present two algorithms that enumerate all minimal stoichiometric precursor sets. The first one is of theoretical interest only and is based on the above mentioned relationship. The second approach solves a series of mixed integer linear programming problems. We compared the computed minimal precursor sets to experimentally obtained growth media of several Escherichia coli strains using genome-scale metabolic networks. Conclusions: The results show that the second approach efficiently enumerates minimal precursor sets taking stoichiometry into account, and allows for broad in silico studies of strains or species interactions that may help to understand e.g. pathotype and niche-specific metabolic capabilities. sasita is written in Java, uses cplex as LP solver and can be downloaded together with all networks and input files used in this paper at http://www.sasita.gforge.inria.fr