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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.
... If P1 is polynomial-time solvable for some class of games, then so are P2 and P3 using the ellipsoid method [19,21]. As the core of a game might be empty, other solution concepts are also considered. ...
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... for a graph G with n vertices, and where the minimum is taken over all ortho-normal representations U of G, meaning sets of real unit-vectors U " u i P R N | 1 ď i ď n, |u i | 2 " 1 ( , where the vertices of G are labeled by i P t1, 2, ..., nu, such that xu i , u j y " 0 if vertices i and j are not connected and c P R N for some N (not necessarily equaling n). ϑpGq can be calculated from a semi-definite program [5], which means that it, unlike ΘpGq, is relatively easy to numerically calculate within an arbitrary small margin of error. An important property of ϑpGq is that it upper bounds ΘpGq, i.e. ...
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