A subexponential lower bound for the Least Recently Considered rule for solving linear programs and games

The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most 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 Cunningham’s Least Recently Considered rule [5], which belongs to the family of history-based rules. Also known as the ROUND-ROBIN rule, Cunningham’s pivoting method fixes an initial ordering on all variables first, and then selects the improving variables in a round-robin fashion. We provide the first subexponential (i.e., of the form 2Ω( √ n)) lower bound for this rule in a concrete setting. Our lower bound is obtained by utilizing connections between pivoting steps performed by simplex-based algorithms and improving switches performed by policy iteration algorithms for 1player and 2-player games. We start by building 2-player parity games (PGs) on which the policy iteration with the ROUND-ROBIN rule performs a subexponential number of iterations. We then transform the parity games into 1-player Markov Decision Processes (MDPs) which correspond almost immediately to concrete linear programs.