Analysis of L-Structure Stability of Convex Integer Programming Problems

  • M. V. Devyaterikova
  • A. A. Kolokolov
Part of the Operations Research Proceedings book series (ORP, volume 2000)


A number of results in integer programming (IP) were obtained on the basis of regular partitions [1,3,4]. These results include bounds on the number of iterations of dual fractional cutting plane and branch and bound algorithms, development of new L-class enumeration algorithms for analysis and solving of IP problems, etc. Analysis of the integer programming problems stability is a new application of the approach [1,4]. Unlike the authors of [2,5,6] and other papers we study do not only the conditions of stability of optimal solutions, but also the stability of relaxation set of the problems. These questions are investigated here for the convex integer programming problems on the basis of L-partition. New properties of L-structure of the relaxation set are obtained. Upper bounds on cardinalities of the L-intervals are constructed under small enough variations of this set. Here we develop further the results presented in [1,4]. A similar bound on cardinality of L-covering (which is a special case of L-interval) has been obtained in [4].


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  1. R. G. Busacker, T. L. Saaty (1965). Finite graphs and networks: an introduction with applications. New York: Mc Graw Hill Book Company. XIV, 294 p.Google Scholar
  2. J. Edmonds, E. L. Johnson (1970) Matching: a well solvable class of integer linear programs. In Combinatorial Structures and their Applications, Gordon and Breach, New York, 89–92.Google Scholar
  3. E. Kh. Gimadi, N. I. Glebov, and V. A. Perepelitsa (1976) Algorithms with estimates for discrete optimization problems (in Russian), in: Problemy Kibernet. Vol. 31, Nauka, Moscow, pp. 35–42.Google Scholar
  4. S. Hakimi (1962). On the readability of a set of integers as degrees of the vertices of a graph. J. SIAM Appl. Mathg., 10, 496–506.CrossRefGoogle Scholar
  5. F. Harary (1969). Graph theory. Addison-Wesley, Reading, Massachusetts.Google Scholar
  6. V. Havel (1955). A note to question of existence of finite graphs. Casopis Pest Mat., 80, 477–480.Google Scholar
  7. A. V. Kostochka, A. I. Serdyukov (1985) Polynomial algorithms with estimates 3/4 5/6 for the maximum-weight traveling salesman problem (in Russian) Upravlyaemye sistemy, 26, 55–59.Google Scholar
  8. A. I. Serdyukov (1991). Polynomial algorithms with estimates of accuracy of solutions for one class of the maximum weight TSP (in Russian). Kombinator.-algebr. metody v diskret, optimiz. Mezhvuzovskii sbornik. N. Novgorod, 107–114.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • M. V. Devyaterikova
    • 1
  • A. A. Kolokolov
    • 1
  1. 1.Omsk Branch of Sobolev Institute of MathematicsSiberian Branch of Russian Academy of SciencesRussia

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