Skip to main content
SpringerLink
Log in

An Integral Simplex Algorithm for Solving Combinatorial Optimization Problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper a local integral simplex algorithm will be described which, starting with the initial tableau of a set partitioning problem, makes pivots using the pivot on one rule until no more such pivots are possible because a local optimum has been found. If the local optimum is also a global optimum the process stops. Otherwise, a global integral simplex algorithm creates and solves the problems in a search tree consisting of a polynomial number of subproblems, subproblems of subproblems, etc. The solution to at least one of these subproblems is guaranteed to be an optimal solution to the original problem. If that solution has a bounded objective then it is an optimal set partitioning solution of the original problem, but if it has an unbounded objective then the original problem has no feasible solution. It will be shown that the total number of pivots required for the global integral simplex method to solve a set partitioning problem having m rows, where m is an arbitrary but fixed positive integer, is bounded by a polynomial function of n.

A method for programming the algorithms in this paper to run on parallel computers is discussed briefly.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Balas and M. Padberg, “On the set covering problem,” Operations Research, vol. 20, pp. 1153–1161, 1972.

    Google Scholar 

  2. E. Balas and M. Padberg, “On the set covering problem II: An algorithm for set partitioning,” Operations Research, vol. 23, pp. 74–90, 1975.

    Google Scholar 

  3. V. Chvatal, Linear Programming, Freeman: San Francisco, CA, 1983.

    Google Scholar 

  4. K.L. Hoffman and M. Padberg, “Solving airline crew scheduling problems by branch-and-cut,” Management Science, vol. 39, pp. 657–682, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Gerald L. Thompson

Authors
  1. Gerald L. Thompson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Thompson, G.L. An Integral Simplex Algorithm for Solving Combinatorial Optimization Problems. Computational Optimization and Applications 22, 351–367 (2002). https://doi.org/10.1023/A:1019758821507

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019758821507

Keywords

Search

Navigation