The ellipsoid method and its consequences in combinatorial optimization

M. Grötschel, L. Lovász, A. Schrijver

Research output: Contribution to journalArticle

1028 Citations (Scopus)

Abstract

L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

Original languageEnglish
Pages (from-to)169-197
Number of pages29
JournalCombinatorica
Volume1
Issue number2
DOIs
Publication statusPublished - Jun 1981

Fingerprint

Ellipsoid Method
Combinatorial optimization
Polynomial Algorithm
Combinatorial Optimization
Fractional Chromatic number
Matroid Intersection
Perfect Graphs
Combinatorial Problems
Nonlinear Optimization
Polynomials
Digraph
Packing
Nonlinear Problem
Linear Inequalities
Covering
NP-complete problem
Optimization Problem
Vertex of a graph

Keywords

  • AMS subject classification (1980): 90 C XX, 05 C XX, 90C25, 90C10

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Mathematics(all)
  • Computational Mathematics

Cite this

The ellipsoid method and its consequences in combinatorial optimization. / Grötschel, M.; Lovász, L.; Schrijver, A.

In: Combinatorica, Vol. 1, No. 2, 06.1981, p. 169-197.

Research output: Contribution to journalArticle

Grötschel, M. ; Lovász, L. ; Schrijver, A. / The ellipsoid method and its consequences in combinatorial optimization. In: Combinatorica. 1981 ; Vol. 1, No. 2. pp. 169-197.
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