Abstract
We study the integer recognition problem on cut polytope relaxations. We establish characteristic properties of relaxation points that preclude efficient solution of the problem. We give examples of such points
Similar content being viewed by others
References
Bondarenko, V.A. and Uryvaev, B.V., On One Problem of Integer Optimization, Autom. Remote Control, 2007, vol. 68, no. 6, pp. 948–953.
Boss, V., Lektsii po matematike, tom 10: Perebor i effektivnye algoritmy (Lectures in Mathematics, vol. 10: Enumeration and Efficient Algorithms), Moscow: LKI, 2008.
Bondarenko, V.A., On One Combinatorial Polytope, in Modeling and Analysis of Computational Systems, Yaroslavl: Yaroslav. Gos. Univ., 1987, pp. 133–134.
Padberg, M.V., The Boolean Quadratic Polytope: Some Characteristics, Facets and Relatives, Math. Program., 1989, vol. 45, pp. 139–172.
Deza, M.M. and Laurent, M., Geometry of Cuts and Metrics (Algorithms and Combinatorics), Berlin: Springer, 2009, 2nd ed.
Bondarenko, V.A. and Nikolaev, A.V., A Class of Hypergraphs and Vertices of Cut Polytope Relaxations, Dokl. Math., 2012, vol. 85, no. 1, pp. 46–47.
Ziegler, G.M., Lectures on 0–1 Polytopes, in Polytopes-Combinatorics and Computation, DMV Seminars Series, Kalai, G. and Ziegler, G.M., Eds., Basel: Birkhauser, 2000.
Nikolaev, A.V., Hypergraphs of a Special Form and Analyzing the Properties of Cut Polytope Relaxations, Modelir. Anal. Inform. Sist., 2011, vol. 18, no. 3, pp. 82–100.
Gawrilow, E. and Joswig, M., Polymake: A Framework for Analyzing Convex Polytopes, in Polytopes-Combinatorics and Computation (Oberwolfach, 1997), DMV Sem., 29, Basel: Birkhauser, 2000, pp. 43–73.
Christof, T. and Loebel, A., PORTA: Polyhedron Representation Transformation Algorithm, Version 1.4.1, The Konrad-Zuse-Zentrum fur Informationstechnik Berlin, http://www.zib.de/Optimization/Software/Porta/.
Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, A Series of Books in the Mathematical Sciences, San Francisco: W.H. Freeman and Co., 1979.
Gebser, M., Kaufmann, B., Neumann, A., and Schaub, T., Conflict-Driven Answer Set Solving, Proc. Twentieth Int. Joint Conf. Artificial Intelligence (IJCAI’07), Hyderabad, India: AAAI Press/MIT Press, 2007, pp. 386–392.
Biere, A. and Kepler, J., Plingeling, Linz: Univ. of Linz, Austria, http://fmv.jku.at/lingeling/.
Berkelaar, M., Eikland, K., and Notebaert, P., lp solve 5.5.2.0. Open Source (Mixed-Integer) Linear Programming System, http://lpsolve.sourceforge.net/5.5/.
Makhorin, A.O., GLPK: GNU Linear Programming Kit 4.53, http://www.gnu.org/software/glpk/.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.A. Bondarenko, A.V. Nikolaev, M.E. Symanovich, R.O. Shemyakin, 2014, published in Avtomatika i Telemekhanika, 2014, No. 9, pp. 108–121.
Rights and permissions
About this article
Cite this article
Bondarenko, V.A., Nikolaev, A.V., Symanovich, M.E. et al. On a recognition problem on cut polytope relaxations. Autom Remote Control 75, 1626–1636 (2014). https://doi.org/10.1134/S0005117914090082
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117914090082