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.
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Research by the third author was supported by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).
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Grötschel, M., Lovász, L. & Schrijver, A. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981). https://doi.org/10.1007/BF02579273
AMS subject classification (1980)
- 90 C XX, 05 C XX
- 90 C 25, 90 C 10