The grammar problem, a generalization of the single-source shortest-path problem introduced by Knuth, is to compute the minimum-cost derivation of a terminal string from each non-terminal of a given context-free grammar, with the cost of a derivation being suitably defined. This problem also subsumes the problem of finding optimal hyperpaths in directed hypergraphs (under varying optimization criteria) that has received attention recently. In this paper we present an incremental algorithm for a version of the grammar problem. As a special case of this algorithm we obtain an efficient incremental algorithm for the single-source shortest-path problem with positive edge lengths. The aspect of our work that distinguishes it from other work on the dynamic shortest-path problem is its ability to handle "multiple heterogeneous modifications": between updates, the input graph is allowed to be restructured by an arbitrary mixture of edge insertions, edge deletions, and edge-length changes.

We deal with directed hypergraphs as a tool to model and solve some classes of problems arising in Operations Research and in Computer Science. Concepts such as connectivity, paths and cuts are defined. An extension of the main duality results to a special class of hypergraphs is presented. Algorithms to perform visits of hypergraphs and to find optimal paths are studied in detail. Some applications arising in propositional logic, And-Or graphs, relational data bases and transportation analysis are presented. January 1990 Revised, October 1992 ( * ) This research has been supported in part by the "Comitato Nazionale Scienza e Tecnologia dell'Informazione", National Research Council of Italy, under Grant n.89.00208.12, and in part by research grants from the National Research Council of Canada. 1 Dipartimento di Informatica, Università di Pisa, Italy 2 Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, Canada 2 INTRODUCTION Hypergraphs, a generaliz...

A directed hypergraph is defined by a set of nodes and a set of hyperarcs, each connecting a set of source nodes to a single target node. Directed hypergraphs are used in several contexts to model different combinatorial structures, such as functional dependencies [Ull82], Horn clauses in propositional calculus [AI91], AND-OR graphs [Nil82], Petri nets [Pet62]. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, hyperpaths are suitable of different definitions of measure, corresponding to different concepts arising in various applications. In this paper we consider the problem of finding minimal hyperpaths according to several measures. We show that some of these problems are, not surprisingly, NP-hard. However, if the measure function on hyperpaths matches certain conditions (which we define as value-based measure functions) , the problem turns out to be solvable in polynomial time. We...

We consider the problem of finding an optimal set of feasible assembly operations in order to derive a desired product from a given set of parts. We show that optimal solutions, under different objective functions, can be obtained by means of computationally efficient hypergraph algorithms. In particular, we consider the case in which one wants to minimize the duration with an unlimited degree of parallelism. Finally, a new fast algorithm is presented in order to assign operations to robots in an optimal way when the degree of parallelism is bounded. Technical Report TR-6/92 March, 1992 (revised: July, 1992) This research has been supported by "Ministero dell'Universit e della Ricerca Scientifica e Tecnologica", National Project on "Gestione dei flussi nei sistemi flessibili di produzione". 2 Introduction In robotic assembly systems, one or more types of items are produced by assembling together several parts given in input, according to a properly defined assembly plan. An asse...

We address a generalization of graphs, the directed hypergraphs, and show that they are a powerful tool in modelling and solving several relevant problems in many application areas. Such application areas include Linear Production Problems, Flexible Manufacturing Systems, Propositional Logic, Relational Databases, and Public Transportation Systems.

. Directed hypergraphs are used in several applications to model different combinatorial structures. A directed hypergraph is defined by a set of nodes and a set of hyperarcs, each connecting a set of source nodes to a single target node. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, however, hyperpaths are suitable of many different definitions of measure, corresponding to different concepts arising in various applications. In this paper we consider the problem of finding optimal hyperpaths according to several measures. We also provide results that may shed some light on the intrinsic complexity of finding optimal hyperpaths. 1 Introduction A directed hypergraph is a generalization of the concept of directed graph. It was first introduced in [2] to represent functional dependencies in relational data base schemata. While directed graphs are normally used for representing one-t...

A partial instantiation approach to the solution of the satisfiability problem in the Schoenfinkel-Bernays fragment of 1 st order logic is presented. It is based on a reduction of the problem to a finite sequence of satisfiability problems in the propositional logic and it improves upon the original idea of partial instantiation, as proposed by Jeroslow. In the second part of the paper a new interpretation of the partial instantiation approach in terms of Directed Hypergraphs is proposed and a particular implementation for the Datalog case is described in detail. 1 Introduction. The problem of Logical Inference plays a fundamental role in Decision Sciences and has several applications in fields such as decision support systems, logic circuit design, data bases, and programming languages. Although classical approaches to formalize and solve inference problems have been of symbolic nature, in the last few years many scientists in the Operations Research community have studied ...

Abstract—In many practical situations, we know the values of some quantities x1,..., xn, we know the relations between these quantities, the desired quantity y, and maybe some auxiliary quantities, and we want to estimate y. There exist automatic tools for such estimations – called program synthesis tools. A program synthesis tool usually generates a program for computing y. In many cases, however, several such programs are possible, and it is desirable to generate the optimal (e.g., the fastest) program. In this paper, we describe algorithms aimed at such optimal program synthesis. The problem can be interpreted in logical terms, as assigning fuzzy-style degrees to rules describing relations between variables. I. NEED FOR DATA PROCESSING: A BRIEF REMINDER One of the main objectives of science: an informal description. One of the main objectives of science is to describe the

In this paper we present a decremental algorithm for maintaining of minimum rank hyperpaths in a directed hypergraph from a source vertex s to all other vertices, under the assumption of unit hyperedge weights. Given a hypergraph H with n vertices and m hyperedges, the total time needed to perform a sequence of m hyperedge deletions is O(n \Delta Size(H)), where Size(H) is the sum of the sizes of the hyperedges of H; the total space needed is O(n + Size(H)). Thus, the amortized time per deletion is O (Size(H) \Delta n=m). In the case of integer hyperedge weights in [1; C] our solution requires O(C \Delta n \Delta Size(H)) total time and O(C + n + Size(H)) space. The only algorithm proposed in the literature for the fully dynamic maintenance of minimum weight hyperpaths may take total time O(n 2 \Delta Size(H)) for a sequence of m deletions. Thus, our algorithm gives an n speedup for the amortized time per deletion. Using the algorithm presented in this paper, we also show how to mai...