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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 Ω(n α ) , for some α > 0) lower bounds for the two most natural, and most studied, randomized pivoting rules suggested to date.The first randomized pivoting rule considered is RandomEdge, 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 Matoušek, Sharir and Welzl. Our lower bound for the Random-Facet pivoting rule essentially matches the subexponential upper bounds given by Kalai and by Matoušek 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).

This paper presents a new lower bound for the discrete strategy improvement algorithm for solving parity games due to Vöge and Jurdziński. First, we informally show which structures are difficult to solve for the algorithm. Second, we outline a family of games of quadratic size on which the algorithm requires exponentially many strategy iterations, answering in the negative the long-standing question whether this algorithm runs in polynomial time. Additionally we note that the same family of games can be used to prove a similar result w.r.t. the strategy improvement variant by Schewe.

ABSTRACT. This paper presents a new exponential lower bound for the two most popular deterministic variants of the strategy improvement algorithms for solving parity, mean payoff, discounted payoff and simple stochastic games. The first variant improves every node in each step maximizing the current valuation locally, whereas the second variant computes the globally optimal improvement in each step. We outline families of games on which both variants require exponentially many strategy iterations.