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Random-Facet and Random-Bland require subexponential time even for shortest paths [article]

Oliver Friedmann and Thomas Dueholm Hansen and Uri Zwick
2014 arXiv   pre-print
The Random-Facet algorithm of Kalai and of Matousek, Sharir and Welzl is an elegant randomized algorithm for solving linear programs and more general LP-type problems.  ...  Its expected subexponential time of $2^{\tilde{O}(\sqrt{m})}$, where $m$ is the number of inequalities, makes it the fastest known combinatorial algorithm for solving linear programs.  ...  of pivoting steps performed by Random-Facet and Random-Facet 1P are not the same.  ... 
arXiv:1410.7530v1 fatcat:pfnzvhm2szhlxjjnzftdryxk3m

The many facets of linear programming

Michael J. Todd
2002 Mathematical programming  
We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods.  ...  It is known that this bound fails for unbounded polyhedra (Klee and Walkup [61] ), and it also fails for bounded polyhedra (polytopes) if the path is required to be monotonic with respect to the objective  ...  the discussion in Bland et al  ... 
doi:10.1007/s101070100261 fatcat:jij5zyvtmbectexvrji4rmzjkm