A probabilistic analysis of the simplex method
A probabilistic analysis of the simplex method
The Hirsch conjecture is true for (0,1)-polytopes
Mathematical Programming: Series A and B
Random walks, totally unimodular matrices, and a randomised dual simplex algorithm
Mathematical Programming: Series A and B
A polynomial time primal network simplex algorithm for minimum cost flows
Mathematical Programming: Series A and B
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Improved smoothed analysis of multiobjective optimization
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On sub-determinants and the diameter of polyhedra
Proceedings of the twenty-eighth annual symposium on Computational geometry
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We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope $P = \left\{ x \in \mathbb{R}^n \,\colon\, Ax \leq b \right\}$ along the edges of P, where A∈ℝm ×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that is a measure for the flatness of the vertices of P. For integer matrices A∈ℤm ×n we show a connection between δ and the largest absolute value Δ of any sub-determinant of A, yielding a bound of O(Δ4mn4) for the length of the computed path. This bound is expressed in the same parameter Δ as the recent non-constructive bound of O(Δ2n4 log(n Δ)) by Bonifas et al. [1]. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(mn4), which significantly improves the previously best known constructive bound of O(m16n3 log3 (mn)) by Dyer and Frieze [7].