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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 nonpolynomial 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 nonpolynomial lower bounds were known, prior to this work, for Cunningham’s Least Recently Considered rule [5], which belongs to the family of historybased rules. Also known as the ROUNDROBIN rule, Cunningham’s pivoting method fixes an initial ordering on all variables first, and then selects the improving variables in a roundrobin 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 simplexbased algorithms and improving switches performed by policy iteration algorithms for 1player and 2player games. We start by building 2player parity games (PGs) on which the policy iteration with the ROUNDROBIN rule performs a subexponential number of iterations. We then transform the parity games into 1player 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 RandomFacet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by RandomFacet∗, a variant of RandomFacet 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 RandomFacet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by RandomFacet∗, a variant of RandomFacet that bases its random decisions on one random permutation. We then obtained a lower bound on the expected number of pivoting steps performed by RandomFacet ∗ and claimed that the same lower bound holds also for RandomFacet. Unfortunately, the claim that the expected number of steps performed by RandomFacet and RandomFacet ∗ 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 mucalculus and modelchecking
"... This chapter presents a part of the theory of the mucalculus that is relevant to the, broadly understood, modelchecking problem. The mucalculus is one of the most important logics in modelchecking. 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 mucalculus that is relevant to the, broadly understood, modelchecking problem. The mucalculus is one of the most important logics in modelchecking. 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 mucalculus. It discusses the theory of the mucalculus starting with the tree model property, and bisimulation invariance. Then it develops the notion of modal automaton: an automatonbased model behind the mucalculus. 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 mucalculus to monadic secondorder logic as well as to some program and temporal logics. It also presents two extensions of the mucalculus that allow us to address issues such as inverse modalities.
An improved version of the RandomFacet pivoting rule for the simplex algorithm
, 2015
"... The RandomFacet 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 RandomFacet 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