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A deterministic subexponential algorithm for solving parity games
 SODA
, 2006
"... The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms ..."
Abstract

Cited by 80 (3 self)
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The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above.
Simple stochastic games and Pmatrix generalized linear complementarity problems
 PROC. 15TH INTERNATIONAL SYMPOSIUM ON FUNDAMENTALS OF COMPUTATION THEORY (FCT
, 2005
"... We show that the problem of finding optimal strategies for both players in a simple stochastic game reduces to the generalized linear complementarity problem (GLCP) with a Pmatrix, a wellstudied problem whose hardness would imply NP = coNP. This makes the rich GLCP theory and numerous existing al ..."
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Cited by 5 (2 self)
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We show that the problem of finding optimal strategies for both players in a simple stochastic game reduces to the generalized linear complementarity problem (GLCP) with a Pmatrix, a wellstudied problem whose hardness would imply NP = coNP. This makes the rich GLCP theory and numerous existing algorithms available for simple stochastic games. As a special case, we get a reduction from binary simple stochastic games to the Pmatrix linear complementarity problem (LCP).
UPSALIENSIS
"... In this thesis we investigate how the known framework of automatic formal verification by model checking can be extended in different directions. One extension is to go beyond the common limitation of the existing specification formalisms, that they can describe only regular properties of components ..."
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In this thesis we investigate how the known framework of automatic formal verification by model checking can be extended in different directions. One extension is to go beyond the common limitation of the existing specification formalisms, that they can describe only regular properties of components. This can be achieved using logics capable of expressing nonregular properties, such as the Propositional Dynamic Logic of Contextfree Programs (PDLCF), Fixpoint Logic with Chop (FLC) or the Higherorder Fixpoint Logic (HFL). Our main result in this area is proving that the problem of model checking HFL formulas of order bounded by k is kEXPTIME complete. In the proofs we demonstrate two model checking algorithms for that logic. We also show that PDLCF is equivalent to a proper fragment of FLC. The standard model checking algorithms, which are run on a single computer, are severely limited by the amount of available computing resources. A way to overcome this limitation is to develop distributed algorithms, which can be run on a cluster of computers and use their joint resources. In this thesis we show how a distributed model checking algorithm for the alternationfree fragment of the modal μcalculus can be extended to handle formulas with one level of alternation. This is an important extension, since Lμ formulas with one level of alternation can
On the complexity of parity games
, 2008
"... Parity games underlie the model checking problem for the modal µcalculus, the complexity of which remains unresolved after more than two decades of intensive research. The community is split into those who believe this problem – which is known to be both in NP and coNP – has a polynomialtime solut ..."
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Parity games underlie the model checking problem for the modal µcalculus, the complexity of which remains unresolved after more than two decades of intensive research. The community is split into those who believe this problem – which is known to be both in NP and coNP – has a polynomialtime solution (without the assumption that P = NP) and those who believe that it does not. (A third, pessimistic, faction believes that the answer to this question will remain unknown in their lifetime.) In this paper we explore the possibility of employing Bounded Arithmetic to resolve this question, motivated by the fact that problems which are both NP and coNP and which can be formulated within a certain fragment of Bounded Arithmetic necessarily admit a polynomialtime solution. While the problem remains unresolved by this paper, we do proposed another approach, and at the very least provide a modest refinement to the complexity of parity games (and in turn the µcalculus model checking problem): that they lie in the class PLS of Polynomial Local Search problems. This result is based on a new proof of memoryless determinacy which can be formalised in Bounded Arithmetic. The approach we propose may offer a route to a polynomialtime solution. Alternatively, there may be scope in devising a reduction of the problem to some other problem which is hard with respect to PLS, thus making the discovery of a polynomialtime solution unlikely according to current wisdom. 1 1