, Volume 1, Issue 2, pp 169–197

The ellipsoid method and its consequences in combinatorial optimization


  • M. Grötschel
    • Universität Bonn, Inst. für Ökonometrie und Operations Research
  • L. Lovász
    • Bolyai InstituteA. József University
  • A. Schrijver
    • Inst. Actuariaat en EconometrieUniversity of Amsterdam

DOI: 10.1007/BF02579273

Cite this article as:
Grötschel, M., Lovász, L. & Schrijver, A. Combinatorica (1981) 1: 169. doi:10.1007/BF02579273


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.

AMS subject classification (1980)

90 C XX, 05 C XX90 C 25, 90 C 10

Copyright information

© Akadémiai Kiadó 1981