, Volume 1, Issue 2, pp 169-197

First online:

The ellipsoid method and its consequences in combinatorial optimization

  • M. GrötschelAffiliated withUniversität Bonn, Inst. für Ökonometrie und Operations Research
  • , L. LovászAffiliated withBolyai Institute, A. József University
  • , A. SchrijverAffiliated withInst. Actuariaat en Econometrie, University of Amsterdam

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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 XX 90 C 25, 90 C 10