Decomposition and parallelization techniques for enumerating the facets of combinatorial polytopes

International Journal of Computational Geometry and Applications

Volumn 11, Issue 4, 2001, Pages 423-437

  Christof, Thomas ^a   Reinelt, Gerhard ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

0 1 polytope; Convex hull; Facet enumeration; Linear description

Indexed keywords

EID: 0035601976     PISSN: 02181959     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218195901000560     Document Type: Article

References (22)

1. T.S. Motzkin, H. Raiffa, G.L. Thompson, and R.M. Thrall, "The double description method," in Contributions to the Theory of Games, volume 2, eds. H.W. Kuhn and A.W. Tucker, (Princeton University Press, Princeton NJ, 1953).
2. T. Christof and G. Reinelt, "Combinatorial optimization and small polytopes," Top 4 (1996) 1-64.
3. K. Fukuda and A. Prodon, "Double description method revisited," Combinatorics and Computer Science, volume 1120 of Lecture Notes in Computer Science, pp. 91-111. (Springer, Heidelberg, 1996).
4. M. Padberg, Linear Optimization and Extensions, volume 12 of Algorithms and Combinatorics. (Springer, Heidelberg, 1995).
5. D. Avis, D. Bremner, and R. Seidel, "How good are convex hull algorithms?," Computational Geometry: Theory and Applications 7 (1997) 265-301.
6. T. Christof, Low-Dimensional 0/1-Polytopes and Branch-and-Cut in Combinatorial Optimization (Shaker, Aachen, 1997).
7. A. Brüngger, A. Marzetta, K. Fukuda, and J. Nievergelt, "The parallel search bench ZRAM and its applications," Proc. Parallel Optimization Colloquium POC, Versailles, France, 1996.
8. D. Avis and K. Fukuda, "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra," Discrete and Computational Geometry 8 (1992) 295-313.
9. D. Avis and K. Fukuda, "Reverse search for enumeration," Discrete Applied Mathematics 65 (1996) 21-46.
10. T. Christof, "SMAPO - library of linear descriptions of low-dimensional 0/1-Polytopes connected with SMAll instances of combinatorial optimization problems," 1995. http://www.iwr.uni-heidelberg.de/iwr/comopt/soft/SMAPO/SMAPO.html.
11. D. Naddef and G. Rinaldi, "The graphical relaxation: A new framework for the symmetric travelling salesman polytope," Mathematical Programming 58 (1993) 53-87.
12. T. Christof, M. Jünger, and G. Reinelt, "A complete description of the traveling salesman polytope on 8 nodes," Operations Research Letters 10 (1991) 497-500.
13. D. Alevras and M. Padberg, Personal communication (1995).
14. G. Rinaldi, Personal communication, 1995.
15. G. Reinelt, The Linear Ordering Problem: Algorithms and Applications, research and exposition in mathematics 8, (Heldermann, Berlin, 1985).
16. G. Bolotashvili, M. Kovalev and E. Girlich, "New facets of the linear ordering polytope," SIAM J. Discrete Math. 12 (1999) 326-336.
17. F. Barahona and R. Mahjoub, "On the cut polytope," Mathematical Programming 36 (1986) 157-173.
18. U. Kortenkamp, J. Richter-Gebert, A. Sarangarajan and G.M. Ziegler. "Extremal Properties of 0/1-Polytopes" Discrete & Computational Geometry 17 (1997) 439-448
19. G. Ziegler, "Recent progress on polytopes", in Advances in Discrete and Computational Geometry, eds. B. Chazelle, J.E. Goodman and R. Pollack, Contemporary Mathematics 223, (Amer. Math. Soc., Providence RI, 1998).
20. I. Bárány and A. Pór, "0-1 polytopes with many facets," Technical Report, Hungary Academy of Sciences, Budapest, 2000.
21. V. Kaibel, T. Fleiner, and G. Rote, "Upper bounds on the maximal number of facets of 0/1-polytopes," European Journal of Combinatorics 21 (2000) 121-130.
22. T. Christof and G. Reinelt, "Algorithmic Aspects of Using Small Instance Relaxations in Parallel Branch-and-Cut," to appear in: Algorithmica (2001).