Subexponential lower bounds for randomized pivoting rules for solving linear programs

The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most deterministic pivoting rules are known, however, to need 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 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 we consider 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 we consider is Random-Facet, a more complicated randomized pivoting rule suggested by Matoušek, Sharir and Welzl [MSW96]. Our lower bound for the Random-Facet pivoting rule essentially matches the subexponential upper bound of Matoušek et al. [MSW96]. 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 1-player and 2-player games. We start by building 2-player parity games (PGs) on which suitable randomized policy iteration algorithms perform a subexponential number of iterations. We then transform these 2-player games into 1-player Markov Decision Processes (MDPs) which correspond almost immediately to concrete linear programs. ∗Department of Computer Science, University of Munich, Germany. E-mail: [email protected]. †Department of Computer Science, Aarhus University, Denmark. Supported by the Center for Algorithmic Game Theory, funded by the Carlsberg Foundation. E-mail: [email protected]. ‡School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Research supported by grant no. 1306/08 of the Israel Science Foundation. E-mail: [email protected].