The Gomory-Chvátal Closure of a Non-Rational Polytope is a Rational Polytope
The question as to whether the Gomory-Chvátal closure of a non-rational polytope is a polytope has been a longstanding open problem in integer programming. In this paper, we answer this question in the affirmative, by combining ideas from polyhedral theory and the geometry of numbers.
KeywordsConvex Body Valid Inequality Integral Vector Rational Polyhedron Polyhedral Theory
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- 1.Chv´atal, V.: Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematics 4, 305–337 (1973)Google Scholar
- 2.Cornu´ejols, G.: Valid inequalities for mixed integer linear programs, Mathematical Programming B 112, 3–44 (2008)Google Scholar
- 3.Dadush, D.; Dey, S. S.; Vielma, J. P.: The Chv´atal-Gomory closure of a strictly convex body, http://www.optimization-online.org/DB/HTML/2010/05/2608.html (2010)
- 4.Dadush, D.; Dey, S. S.; Vielma, J. P.: The Chv´atal-Gomory closure of a compact convex set, Proceedings of the 15th Conference on Integer Programming and Combinatorial Optimization (IPCO 2011), Lecture Notes in Computer Science Vol. 6655 (O. G¨unl¨uk and G. J.Woeginger, editors), 130–142 (2011)Google Scholar
- 5.Dey, S. S.; Vielma, J. P.: The Chv´atal-Gomory closure of an ellipsoid is a polyhedron, Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization (IPCO 2010), Lecture Notes in Computer Science Vol. 6080 (F. Eisenbrand and F. B. Shepherd, editors), 327–340 (2010)Google Scholar
- 6.Dunkel, J.: The Gomory-Chv´atal Closure: Polyhedrality, Complexity, and Extensions, PhD Thesis, Massachusetts Institute of Technology, http://stuff.mit.edu/people/juliane/thesis/(2011)
- 7.Dunkel, J., Schulz, A. S. The Gomory-Chv´atal Closure of a non-rational polytope is a rational polytope, http://www.optimization-online.org/DBHTML/2010/11/2803.html (2010)
- 11.Schrijver, A.: Theory of Linear and Integer Programming, John Wiley & Sons, Chichester (1986)Google Scholar