Skip to main content
Log in

On 2-neighborly polytopes and the gale construction

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

The Gale construction of 2-neighborly polytopes is investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Belov, Yu.A., About the Density of the Graph Matroid, Modeli issledovaniya operatsii v vychislitel’nykh sistemakh (Yaroslavl),1985, pp. 95–100.

  2. Beloshevskii, M.I., Polytope of the Problem of the Maximum Cut, Modeli i algoritmy matematicheskogo obespecheniya EVM (Yaroslavl), 1986. pp. 12–20.

  3. Bondarenko, V.A., Neighboring Vertices of Polytopes, Generated by Matroid, Voprosy teorii grupp i gomologuicheskoy algebry (Yaroslavl), 1982, pp. 66–68.

  4. Bondarenko, V.A., Non-Polynomial Lower Estimates of the Complexity of the Traveling Salesman Problem in a Some Class of Algorithms, Avtom. Telemekh., 1983, no. 9, pp. 45–50.

  5. Bondarenko, V.A., and Shunikova, E.V., Generalized Permutation Polytopes and Properties of Sorting Algorithms, Modeli issledovaniya operatsii v vychislitel’nykh sistemakh (Yaroslavl), 1985, pp. 105–113.

  6. Bondarenko, V.A., Nonpolynomial Lower Estimates of the Complexity of Optimization Problems about the Clique and Vertex Covering in the Class of Direct Algorithms, Kombinatorno-algebraicheskie metody v prikladnoy matematike (Gor’kiy), 1985, pp. 59–65.

  7. Bondarenko, V.A., Exponentiality of the Density of the Graph of the Polytope of the Three-Combined Problem, Tezisy 7-oy Vsesoyuznoy konferencii “Problemy teoreticheskoy kibernetiki” (Abstracts of the 7th All-Union Conference “Problems of Theoretical Cybernetics”), Irkutsk, 1985. p. 29.

  8. Bondarenko, V.A., Estimates of the Complexity of Combinatorial Optimization Problems in a Some Class of Algorithms, Dokl. Akad. Nauk USSR, 1993, vol. 328, no. 1, pp. 22–24.

    MathSciNet  Google Scholar 

  9. Bondarenko, V.A. Poliedral’nye grafy i slozhnost’ v kombinatornoi optimizatsii (Polyhedral Graphs and the Complexity in Combinatorial Optimization), Yaroslavl: Yaroslavl State University, 1995.

    Google Scholar 

  10. Bondarenko, V.A., Polyhedral Graphs and the Complexity of Combinatorial Optimization Problems, Matematika v Yaroslavskom universitete (Yaroslavl), 1995, pp. 31–42.

  11. Bondarenko, V.A., and Brodsky, A.G., On Random 2-Neighborly 0/1-Polytopes, Diskretnaya matematika, 2008, vol. 20, no. 1, pp. 64–69.

    Google Scholar 

  12. Bondarenko, V.A., Maksimenko, A.N., Geometricheskie construktsii i slozhnost’ v kombinatornoi optimizatsii (Geometric Constructions and Complexity in Combinatorial Optimization), Moscow: Izdatel’stvo LKI, 2008.

    Google Scholar 

  13. Bronsted, A. Vvedenie v teoriyu vypuklykh mnogogrannikov (Introduction to the Theory of Convex Polytopes), Moscow: Mir, 1988.

    Google Scholar 

  14. Greshnev, S.N., Polyhedron of the Problem of m-Vertex Subgraph of a Complete Graph, Zh. Vych. Mat. Mat. Fiz., 1984, vol. 24, no. 5, pp. 790–793.

    MATH  MathSciNet  Google Scholar 

  15. Emelichev, V.A., Kovalev, M.M., and Kravtsov, M.K., Mnogogranniki, graphy, optimizatsiya (combinatornaya teoriya mnogogrannikov) (Polytopes, Graphs, Optimization (Combinatorial Theory of Polytopes), Moscow: Nauka, 1981.

    Google Scholar 

  16. Maksimenko, A.N., A Lower Estimation on the Density of the Graph of the Polyhedron of the Problem of Matching, Sovremennye problemy matematiki i informatiki, 2001, no. 4, pp. 157–161.

  17. Maksimenko, A.N., Reducibility of Discrete Optimization Problems and the Ratio of the Densities of Their Polyhedral Graphs, Model. Anal. Inf. Sist., 2003, vol. 10, no. 2, pp. 3–10.

    MathSciNet  Google Scholar 

  18. Maksimenko, A.N., Polyhedron of the Knapsack Problem, Voprosy teorii grupp i gomologuicheskoy algebry (Yaroslavl), 2003, pp. 163–172.

  19. Maksimenko, A.N., Combinatorial Properties of the Polyhedron of the Problem of Shortest Path, Zh. Vych. Mat. Mat. Fiz., 2004, vol. 44, no. 9, pp. 1693–1696.

    MATH  MathSciNet  Google Scholar 

  20. Barahona, F., and Mahjoub, A.R., On the Cut Polytope, Math. Prog., 1986, vol. 36, pp. 157–173.

    Article  MATH  MathSciNet  Google Scholar 

  21. Caratheodory, C., Ueber den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann, 1907, vol. 1964, pp. 12–17.

    MathSciNet  Google Scholar 

  22. Gale, D., Neighboring Vertices on a Convex Polyhedron, Linear Inequalities and Related Systems, H.W. Kuhn, A.W. Tucker, Eds., Princeton, New Jersey: Princeton University Press, 1956. There is a Russian translation: Gale, D., Sosednie vershiny na vypuklom mnogogrannike, Lineynye neravenstva i smezhnye voprosy (Linear Inequalities and Related Systems), G.U. Kuhn and A.U. Tucker., Eds., Moscow: Inostrannaya Literatura, 1959, pp. 355–362.

    Google Scholar 

  23. Ziegler, G.M., Lectures on Polytopes, NY: Springer, 1995.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. G. Brodskiy.

Additional information

Original Russian Text © A.G. Brodskiy, 2009, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2009, No. 2, pp. 5–21.

About this article

Cite this article

Brodskiy, A.G. On 2-neighborly polytopes and the gale construction. Aut. Conrol Comp. Sci. 44, 434–446 (2010). https://doi.org/10.3103/S0146411610070102

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411610070102

Keywords

Navigation