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By G. Ausiello

Combinatorial difficulties were from the very starting a part of the historical past of arithmetic. via the Sixties, the most periods of combinatorial difficulties were outlined. in the course of that decade, a good number of study contributions in graph thought were produced, which laid the principles for many of the learn in graph optimization within the following years. in the course of the Seventies, a good number of exact goal types have been constructed. The outstanding development of this box considering has been strongly made up our minds by means of the call for of functions and prompted through the technological raises in computing energy and the supply of knowledge and software program. the provision of such simple instruments has ended in the feasibility of the precise or good approximate resolution of enormous scale real looking combinatorial optimization difficulties and has created a few new combinatorial difficulties.

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Let us consider the hypergraph H1, H2 and H3 of Fig. 3. Hz is an SME-hypergraph of H I but it is not strongly equivalent to H I . H, isan SMSE-hypergraph of H. Let us now consider the hypergraphs H and H’ of Fig. 7. The hypergraph H’is both an SME-hypergraph and a SMSE-hypergraph of H. Let us now consider the problem of finding an SME-hypergraph of a hypergraph. Fact 2. An SME-hypergraph of a hypergraph H = can be found in O(IH1’) time. Proof. Let us consider the following algorithm: a) find the closure of all source sets in H and all strong components of H, b) eliminate superfluous source sets, c) eliminate redundant hyperarcs, d) eliminate redundant nodes in all source sets in H.

See [8]page 134. Galil, 2.. “A new Algorithm for the Maximal Flow Problem”, Proc. 9th Ann. Symp. on Foundations of Computers Science, 1978, 231-245. A local-ratio theorem 45 Hochbaum, D. S. “Approximation Algorithms for the Set Covering and Vertex Cover Problems, “ S A M J. Cornput.. vol. 1 1 , 3, 1982, 555-556. Hochbaum. D. App. Math. vol. 6 , 1983, 243-254. Harary. , Graph theory, Addision-Wesly, 1972 (revised). Hopcroft, J . , and R. M. ”SIAM J. , 2, 1973, 225-231. Itai, A,, and M. Rodeh, “Finding a Minimum Circuit in a Graph”, SIAM J.

Therefore, it is natural to look for efficient approximation algorithms. Let A be an approximation algorithm. For graph G with weight functions w , let C, , C* be the cover A produces and an optimum cover, f All log bases, in this paper, me 2. 28 R Bar- Yehuda and S, Even respectively . Define w fcA) R A (G, W ) = and let the performance ratio r A f n ) be w (C*) rA(n) = Sup [ R A ~ G w , ) I G = f V, E l where n = I VI] Many approximation algorithms with performance ratio I 2 have been suggested; see, for example Table 1 .

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