# Two strategies for solving the vertex cover problem on a transputer network

## Abstract

In this article, we present an implementation of a distributed branch and bound algorithm solving the Vertex Cover problem on a network of up to 63 Transputers.

We implemented two different strategies: The first parallelization of our branch and bound algorithm is fully distributed. Every processor performs the same algorithm but on a different part of the solution tree. In this case it is necessary to distribute subproblems among the processors to achieve a well balanced workload.

Our second strategy is based on a tree structured network, where all subproblems are stored at the root processor and the other processes work as slaves of this master process.

To show the performance of our strategies, we solved the Vertex Cover problem for graphs of up to 150 nodes and an average degree of 30. We were able to achieve a speedup of 57.35 for the first strategy on a 60 processor network and 62.11 for the second strategy on 63 processors, compared to a very efficient sequential algorithm.

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## References

- [1]E. Altmann, T. A. Marsland, T. Breitkreutz
*Accounting for Parallel Tree Search Overheads*, Proceedings of the International Conference on Parallel Processing 1988, pp. 198–201Google Scholar - [2]C. Beilken, F. Mattern, M. Reinfrank
*Verteilte Terminierung ein wesentlicher Aspekt der Kontrolle in verteilten Systemen*Sonderforschungsbereich 124 ”VLSI Entwurfsmethoden und Parallelität”, Bericht Nr. 41/85, Dezember 1985Google Scholar - [3]C. Berge
*The theory of graphs and its applications*, Methuen, London 1962Google Scholar - [4]A. Burns
*Programming in OCCAM 2*, Addison Wesely 1988.Google Scholar - [5]R. Feldmann, B. Monien, P. Mysliwietz, O. Vornberger
*Distributed Game Tree Search*, to appear in: Kanal, Gopalakrishnan, Kumar, Parallel Algorithms for Machine Intelligence and Pattern Recognition, North Holland/ Elsevier Publ. Co.Google Scholar - [6]M. R. Garey, D.S. Johnson
*Computers and Intractability: A Guide to the Theory of NP-Completeness*, 1979 Freeman, San Francisco, Calif.Google Scholar - [7]V. K. Janakiram, D. P. Agrawal, R. Mehrotra
*A randomized Parallel Branch and Bound Algorithm*, Proceedings of the International Conference on Parallel Processing 1988, pp. 69–75Google Scholar - [8]R. M. Karp, Y. Zhang,
*A randomized Parallel Branch and Bound Procedure*, Proceedings of the ACM Symposium on Theory of Computing 1988, pp. 290–300Google Scholar - [9]V. Kumar, V. Nageshwara Rao, K. Ramesh
*Parallel Depth First Search on the Ring Architecture*International Conference on Parallel Processing, pp. 128–132Google Scholar - [10]E. L. Lawler, D. E. Wood
*Branch and Bound Methods: A survey*, Operations Research 14, 1966, pp. 699–719Google Scholar - [11]F. C. H. Lin, R. M. Keller
*The Gradient Model Load Balancing Method*, IEEE Transactions on Software Engineering, Vol. 13, No. 1 January 1987Google Scholar - [12]B. Monien and O. Vornberger
*Parallel processing of combinatorial search trees*, Processings International Workshop on Parallel Algorithms and Architectures, Math. Research Nr. 38, Akademie — Verlag Berlin, pp. 60–69, 1987Google Scholar - [13]B. Monien, E. Speckenmeyer, O. Vornberger
*Upperbound for covering problems*, Methods of operations research, 43, 1981, pp. 419–431Google Scholar - [14]R. E. Tarjan, A. E. Trojanowski
*Finding a maximum independent set*, SIAM J. Computing, Vol. 6, No. 3, September 1977, pp. 537–546Google Scholar - [15]O. Vornberger and B. Monien
*Parallel alpha-beta versus parallel SSS**, Proceedings IFIP Conference on Distributed Processing, Distributed Processing, North Holland, pp. 613–625, 1987Google Scholar - [16]O. Vornberger
*Implementing branch and bound in a ring of processors*, Proceedings of CONPAR 86, Lecture Notes of Computer Science 237, Springer Verlag, pp. 157–164, 1986Google Scholar - [17]O. Vornberger
*Load Balancing in a Network of Transputers*, Distributed Algorithms 1987, Lecture Notes of Computer Science 312, Springer Verlag, pp. 116–126Google Scholar