Decomposition and Parallelization Techniques for Enumerating the Facets of Combinatorial Polytopes

A convex polytope can either be described as convex hull of vertices or as solution set of a finite number of linear inequalities and equations. Whereas both representations are equivalent from a theoretical point of view, they are not when optimization problems over the polytope have to be solved. It is a challenging task to convert one description into the other. In this paper we address the efficient computation of the facet structure of several polytopes associated with combinatorial optimization problems. New results are presented which are of interest for theoretical investigations as well as for practical optimization.

[1]  Volker Kaibel,et al.  Simple 0/1-Polytopes , 2000, Eur. J. Comb..

[2]  Günter Rote,et al.  Upper Bounds on the Maximal Number of Facets of 0/1-Polytopes , 2000, Eur. J. Comb..

[3]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[4]  Komei Fukuda,et al.  Double Description Method Revisited , 1995, Combinatorics and Computer Science.

[5]  David Avis,et al.  A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra , 1991, SCG '91.

[6]  Günter M. Ziegler,et al.  Extremal Properties of 0/1-Polytopes , 1997, Discret. Comput. Geom..

[7]  Eberhard Girlich,et al.  New Facets of the Linear Ordering Polytope , 1999, SIAM J. Discret. Math..

[8]  Gerhard Reinelt,et al.  Algorithmic Aspects of Using Small Instance Relaxations in Parallel Branch-and-Cut , 2001, Algorithmica.

[9]  P. McMullen.  THE LINEAR ORDERING PROBLEM: ALGORITHMS AND APPLICATIONS (Research and Exposition in Mathematics 8) , 1987 .

[10]  David Avis,et al.  How good are convex hull algorithms? , 1995, SCG '95.

[11]  G. Reinelt,et al.  Combinatorial optimization and small polytopes , 1996 .

[12]  Jürg Nievergelt,et al.  The parallel search bench ZRAM and its applications , 1999, Ann. Oper. Res..

[13]  Gnter M. Ziegler.  Recent progress on polytopes , 1998 .

[14]  Attila Pór,et al.  On 0-1 Polytopes with Many Facets , 2001 .

[15]  M. Padberg.  Linear Optimization and Extensions , 1995 .

[16]  Giovanni Rinaldi,et al.  The graphical relaxation: A new framework for the symmetric traveling salesman polytope , 1993, Math. Program..

[17]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

[18]  Michael Jünger,et al.  A complete description of the traveling salesman polytope on 8 nodes , 1991, Oper. Res. Lett..