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Approximate minmax relations for odd cycles in planar graphs
 Math Program. Ser. B
"... Abstract. We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertexdisjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss [7] recently proved ..."
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Abstract. We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertexdisjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss [7] recently proved that this ratio is at most 2; we also give a short proof of their result. 1
On largest volume simplices and subdeterminants
, 2014
"... We show that the problem of finding the simplex of largest volume in the convex hull of n points in Qd can be approximated with a factor of O(log d)d/2 in polynomial time. This improves upon the previously best known approximation guarantee of d(d−1)/2 by Khachiyan. On the other hand, we show that t ..."
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We show that the problem of finding the simplex of largest volume in the convex hull of n points in Qd can be approximated with a factor of O(log d)d/2 in polynomial time. This improves upon the previously best known approximation guarantee of d(d−1)/2 by Khachiyan. On the other hand, we show that there exists a constant c> 1 such that this problem cannot be approximated with a factor of cd, unless P = NP. Our hardness result holds even if n = O(d), in which case there exists a c ̄ dapproximation algorithm that relies on recent sampling techniques, where c ̄ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d × n matrix.