Subexponential Lower Bounds For Randomized Pivoting Rules For The Simplex Algorithm

The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. With essentially all deterministic pivoting rules it is known, however, to require an exponential number of steps to solve some linear programs. No non-polynomial lower bounds were known, prior to this work, for randomized pivoting rules. We provide the first subexponential (i.e., of the form 2(Omega(n alpha)), for some alpha > 0) lower bounds for the two most natural, and most studied, randomized pivoting rules suggested to date.The first randomized pivoting rule considered is RANDOM-EDGE, which among all improving pivoting steps (or edges) from the current basic feasible solution (or vertex) chooses one uniformly at random. The second randomized pivoting rule considered is RANDOM-FACET, a more complicated randomized pivoting rule suggested by Kalai and by Matousek, Sharir and Welzl. Our lower bound for the RANDOM-FACET pivoting rule essentially matches the subexponential upper bounds given by Kalai and by Matousek et al. Lower bounds for RANDOM-EDGE and RANDOM-FACET were known before only in abstract settings, and not for concrete linear programs.Our lower bounds are obtained by utilizing connections between pivoting steps performed by simplex-based algorithms and improving switches performed by policy iteration algorithms for solving Markov Decision Processes (MDPs).