Mathematical Programming: Series A and B
A subexponential randomized simplex algorithm (extended abstract)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
The worst-case running time of the random simplex algorithm is exponential in the height
Information Processing Letters
Linear programming, the simplex algorithm and simple polytopes
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
Dynamic Programming and Optimal Control
Dynamic Programming and Optimal Control
Finite State Markovian Decision Processes
Finite State Markovian Decision Processes
A Discrete Strategy Improvement Algorithm for Solving Parity Games
CAV '00 Proceedings of the 12th International Conference on Computer Aided Verification
Automata logics, and infinite games: a guide to current research
Automata logics, and infinite games: a guide to current research
Random Structures & Algorithms
An Exponential Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it
LICS '09 Proceedings of the 2009 24th Annual IEEE Symposium on Logic In Computer Science
Exponential lower bounds for policy iteration
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
Subexponential lower bounds for randomized pivoting rules for the simplex algorithm
Proceedings of the forty-third annual ACM symposium on Theory of computing
A subexponential lower bound for the random facet algorithm for parity games
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Subexponential lower bounds for randomized pivoting rules for the simplex algorithm
Proceedings of the forty-third annual ACM symposium on Theory of computing
Polynomial-Time algorithms for energy games with special weight structures
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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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 Zadeh's pivoting rule [25]. Also known as the Least-Entered rule, Zadeh's pivoting method belongs to the family of memorizing improvement rules, which among all improving pivoting steps from the current basic feasible solution (or vertex) chooses one which has been entered least often. We provide the first subexponential (i.e., of the form 2Ω(√n) lower bound for this rule. 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 1-player and 2-player games. We start by building 2-player parity games (PGs) on which the policy iteration with the LEAST-ENTERED rule performs a subexponential number of iterations. We then transform the parity games into 1-player Markov Decision Processes (MDPs) which corresponds almost immediately to concrete linear programs.