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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
Parity games and propositional proofs
 ACM Transactions on Computational Logic
"... A propositional proof system is weakly automatizable if there is a polynomial time algorithm which separates satisfiable formulas from formulas which have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then ..."
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A propositional proof system is weakly automatizable if there is a polynomial time algorithm which separates satisfiable formulas from formulas which have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time. We give simple proofs that the same holds for depth1 propositional calculus (where resolution has depth 0) with respect to mean payoff and simple stochastic games. We define a new type of combinatorial game and prove that resolution is weakly automatizable if and only if one can separate, by a set decidable in polynomial time, the games in which the first player has a positional winning strategy from the games in which the second player has a positional winning strategy. Our main technique is to show that a suitable weak bounded arithmetic theory proves that both players in a game cannot simultaneously have a winning strategy, and then to translate this proof into propositional form. ∗This research was partially done while the authors were visiting fellows at the Isaac Newton Institute for the Mathematical Sciences in the programme “Semantics & Syntax”. †Partially supported by grant IAA100190902 of GA AV ČR
A Subexponential Lower Bound for Policy Iteration Based on Snare Memorization
, 2011
"... This paper presents a subexponential lower bound for the recently proposed snare memorization nonoblivious strategy iteration algorithm due to Fearnley. Snare memorization is a method to train policy iteration techniques to remember certain profitable substrategies and reapply them again. We show t ..."
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This paper presents a subexponential lower bound for the recently proposed snare memorization nonoblivious strategy iteration algorithm due to Fearnley. Snare memorization is a method to train policy iteration techniques to remember certain profitable substrategies and reapply them again. We show that there is not much hope to find a polynomialtime algorithm for solving parity games by applying such nonoblivious techniques.
The complexity of ergodic meanpayoff games
 In Automata, Languages, and Programming  41st International Colloquium, ICALP 2014
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Symmetric Strategy Improvement
, 2015
"... Symmetry is inherent in the definition of most of the twoplayer zerosum games, including parity, meanpayoff, and discountedpayoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where ..."
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Symmetry is inherent in the definition of most of the twoplayer zerosum games, including parity, meanpayoff, and discountedpayoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann’s traps, which shook the belief in the potential of classic strategy improvement to be polynomial.
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.
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"... Noname manuscript No. (will be inserted by the editor) An exponential lower bound for Cunningham’s rule ..."
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Noname manuscript No. (will be inserted by the editor) An exponential lower bound for Cunningham’s rule