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A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games
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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 ..."
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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 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 2 Ω( √ 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 1-player 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. 1
Errata for: A subexponential lower bound for the Random Facet algorithm for Parity Games
, 2014
"... In [Friedmann, Hansen, and Zwick (2011)] and we claimed that the expected number of pivoting steps performed by the Random-Facet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by Random-Facet∗, a variant of Random-Facet that bases i ..."
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In [Friedmann, Hansen, and Zwick (2011)] and we claimed that the expected number of pivoting steps performed by the Random-Facet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by Random-Facet∗, a variant of Random-Facet that bases its random decisions on one random permutation. We then obtained a lower bound on the expected number of pivoting steps performed by Random-Facet ∗ and claimed that the same lower bound holds also for Random-Facet. Unfortunately, the claim that the expected number of steps performed by Random-Facet and Random-Facet ∗ are the same is false. We provide here simple examples that show that the expected number of steps performed by the two algorithms is not the same. 1
The mu-calculus and model-checking
"... This chapter presents a part of the theory of the mu-calculus that is relevant to the, broadly understood, model-checking problem. The mu-calculus is one of the most important logics in model-checking. It is a logic with an exceptional balance between expressiveness and algorithmic properties. The ..."
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This chapter presents a part of the theory of the mu-calculus that is relevant to the, broadly understood, model-checking problem. The mu-calculus is one of the most important logics in model-checking. It is a logic with an exceptional balance between expressiveness and algorithmic properties. The chapter describes in length the game characterization of the semantics of the mu-calculus. It discusses the theory of the mu-calculus starting with the tree model property, and bisimulation invariance. Then it develops the notion of modal automaton: an automaton-based model behind the mu-calculus. It gives a quite detailed explanation of the satisfiability algorithm, followed by the results on alternation hierarchy, proof systems, and interpolation. Finally, the chapter discusses the relations of the mu-calculus to monadic second-order logic as well as to some program and temporal logics. It also presents two extensions of the mu-calculus that allow us to address issues such as inverse modalities.
An improved version of the Random-Facet pivoting rule for the simplex algorithm
, 2015
"... The Random-Facet pivoting rule of Kalai and of Matoušek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using th ..."
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The Random-Facet pivoting rule of Kalai and of Matoušek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using this rule, on any linear program involving n inequalities in d variables, is
