Dimitris J. Kavvadias

On Horn Envelopes and Hypergraph Transversals

We study the problem of bounding from above and below a given set of bit vectors by the set of satisfying truth assignments of a Horn formula. We point out a rather unexpected connection between the upper bounding problem and the problem of generating all transversals of a hypergraph, and settle several related complexity questions.

An Efficient Algorithm for the Transversal Hypergraph Generation

The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving this problem. We show that the proposed algorithm operates in a way that rules out regeneration and, thus, its memory requirements are polynomially bounded to the size of the input hypergraph. Although no time bound for the algorithm is given, experimental evaluation and comparison with other approaches have shown that it behaves well in practice and it can successfully handle large problem instances.

A linear programming approach to reasoning about probabilities

We continue our study, initiated in [9], of the following computational problem proposed by Nilsson: Several clauses (Boolean functions of several variables) are given, and for each clause the probability that the clause is true is specified. We are asked whether these probabilities are consistent. They are if there is a probability distribution on the truth assignments such that the probability of each clause is the measure of its satisfying set of assignments. Since this is a generalization of the satisfiability problem of predicate calculus, it is immediately NP-hard. In [9] we showed certain restricted cases of the problem to be NP-complete, and used the Ellipsoid Algorithm to show that a certain special case is in P. In this paper we use the Simplex method, column generation techniques, and variable-depth local search to derive an effective heuristic for the general problem. Experiments show that our heuristic performs successfully on instances with many dozens of variables and clauses. We also prove several interesting complexity results that answer open questions in [9] and motivate our approach.

Generating all maximal models of a Boolean expression

We examine the computational problem of generating all maximal models of a Boolean expression in CNF. We give a resolution-like method that reduces the unnegated variables of an expression while preserving its set of maximal models. We present an output-polynomial algorithm for the 2CNF case and we show that the problem cannot be solved in outputpolynomial time in the case of Horn expressions, unless PD NP, despite an affinity of this case to the recently subexponentially solved transversal hypergraph problem. The problem is of course trivial for 1-valid and anti-Horn expressions, and open for exclusive-ors; it is NP-hard in all other cases.

Evaluation of an Algorithm for the Transversal Hypergraph Problem

The Transversal Hypergraph Problem is the problem of computing, given a hypergraph, the set of its minimal transversals, i.e. the hypergraph whose hyperedges are all minimal hitting sets of the given one. This problem turns out to be central in various fields of Computer Science. We present and experimentally evaluate a heuristic algorithm for the problem, which seems able to handle large instances and also possesses some nice features especially desirable in problems with large output such as the Transversal Hypergraph Problem.

Monotone boolean dualization is in co-NP[log 2 n]

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(log n) 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(log2 n) nondeterministic guesses, thus placing the problem in the class co-NP[log2 n]. Our nondeterministic version has also the advantage of having a simpler analysis than the deterministic one.

Locating and Computing All the Simple Roots and Extrema of a Function

This paper describes and analyzes two algorithms for locating and computing with certainty all the simple roots of a twice continuously differentiable function

Hammock-on-ears decomposition: a technique for the efficient parallel solution of shortest paths and other problems

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Efficient Sequential and Parallel Algorithms for the Negative Cycle Problem

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