By G. Ausiello

Combinatorial difficulties were from the very starting a part of the heritage of arithmetic. by means of the Sixties, the most periods of combinatorial difficulties were outlined. in the course of that decade, a good number of study contributions in graph idea have been produced, which laid the principles for many of the examine in graph optimization within the following years. throughout the Seventies, loads of exact function types have been constructed. The awesome development of this box considering that has been strongly decided through the call for of purposes and encouraged through the technological raises in computing energy and the provision of knowledge and software program. the provision of such simple instruments has ended in the feasibility of the precise or good approximate answer of huge scale real looking combinatorial optimization difficulties and has created a few new combinatorial difficulties.

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Figure 7. 5. Conclusions In this paper the problem of determining minimal representations of equivalent directed hypergraphs has been considered. In particular the notion of kernel of a hypergraph has been introduced and a new concept of strong equivalence among hypergraphs has been studied, based on the equivalence of the corresponding kernels. The problem of determining a strongly equivalent hypergraph with the minimum number of source set has been analyzed. This problem, which has relevant applications in relational database research, has been shown to be solvable in polynomial time while other minimal representation problems for directed hypergraphs have been shown to be intractable in a previous iaper 121.

Bertolazzi An example of a mesh and an orthogonal trees for p1 = 4 and p2 = 4 is given in fig, I . In the following we assume that our architectures have an unlimited number of processors to simplify the description of the two procedures; the results in the case of limited number of processors, easily deducible from the results in the case of unlimited number, are also given. Decomposition into modules We give the decomposition into modules in the case of unlimited parallelism, that is in the mesh with P = L processors and in an orthogonal trees with L * L or L * M processors.

Problem is the so called iterative functional procedure (2). In this method the set of admissible decisions U and the set of admissible states S are quantized into a Dynamic pmgramming parallel procedures 51 finite number of values. The set U then consists of the elements u = u , , u2, . . UM and the set S of the elements s = s , , S2’ . . . sL Notation: let us call ue/l the value of the decision variable Sh the value of the present state Sk the value of the next state The iterative functional procedure is given here in natural language: for each stage j, j = N, .