Exact algorithms for solving stochastic games

Hansen, Kristoffer Arnsfelt; Koucký, Michal; Lauritzen, Niels; Miltersen, Peter Bro; Tsigaridas, Elias

doi:10.1145/1993636.1993665

Cited by 43 publications

(35 citation statements)

References 14 publications

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“…The exact constants in the exponents can be useful in many applications e.g. [1,7,8]. Such bounds are also needed to establish a connection between Turing machines and the Blum-Cucker-ShubSmale model and to certify and analyze the Boolean complexity of algorithms based on homotopy techniques [3].…”

Section: Our Resultsmentioning

confidence: 99%

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Bounds for the Condition Number of Polynomials Systems with Integer Coefficients

Herman

Tsigaridas

2015

Lecture Notes in Computer Science

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Abstract. Polynomial systems of equations are a central object of study in computer algebra. Among the many existing algorithms for solving polynomial systems, perhaps the most successful numerical ones are the homotopy algorithms. The number of operations that these algorithms perform depends on the condition number of the roots of the polynomial system. Roughly speaking the condition number expresses the sensitivity of the roots with respect to small perturbation of the input coefficients. A natural question to ask is how can we bound, in the worst case, the condition number when the input polynomials have integer coefficients? We address this problem and we provide effective bounds that depend on the number of variables, the degree and the maximum coefficient bitsize of the input polynomials. Such bounds allows to estimate the bit complexity of the algorithms that depend on the separation bound, like the homotopy algorithms, for solving polynomial systems.

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Section: Our Resultsmentioning

confidence: 99%

“…For ζ * it holds that H(ζ * ) ≤ H(ζ) and so we can use the bound from (7). Putting all these together we have the bound…”

Section: Condition Number For Polynomial Systemsmentioning

confidence: 99%

Bounds for the Condition Number of Polynomials Systems with Integer Coefficients

Herman

Tsigaridas

2015

Lecture Notes in Computer Science

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“…In (B 1 (q * 1 ) − B 1 ( 1 2 (q * 1 + q * 2 )))v this (i, j)'th entry is multiplied by a coordinate of v, which is at least v min . Thus, combining inequalities (13) and (14),…”

Section: A Background Lemmas and Missing Proofsmentioning

confidence: 96%

Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

Stewart

Etessami

Yannakakis

2015

J. ACM

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A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multi-type branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of termination probabilities, and computing these probabilities in turn boils down to computing the least fixed point (LFP) solution of a corresponding monotone polynomial system (MPS) of equations, denoted x = P (x). It was shown by Etessami and Yannakakis [11] that a decomposed variant of Newton's method converges monotonically to the LFP solution for any MPS that has a non-negative solution. Subsequently, Esparza, Kiefer, and Luttenberger [7] obtained upper bounds on the convergence rate of Newton's method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs ([10, 9]). However, prior to this paper, for arbitrary (not necessarily stronglyconnected) MPSs, no upper bounds at all were known on the convergence rate of Newton's method as a function of the encoding size |P | of the input MPS, x = P (x). In this paper we provide worst-case upper bounds, as a function of both the input encoding size |P |, and ǫ > 0, on the number of iterations required for decomposed Newton's method (even with rounding) to converge to within additive error ǫ > 0 of q * , for an arbitrary MPS with LFP solution q * . Our upper bounds are essentially optimal in terms of several important parameters of the problem. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within arbitrary desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) ω-regular or LTL property. ⋆ A full version of this paper is available at arxiv.org/abs/1302.3741.

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“…Shapley showed that these games are determined, meaning that there exists a value vector v, where v s is the value of the game starting at state s. A polynomial-time algorithm has been devised for computing the value vector of a Shapley game when the number of states N is constant [24]. However, since the values may be irrational, this algorithm needs to deal with algebraic numbers, and the degree of the polynomial is O(N ) N 2 , so if N is even mildly super-constant, then the algorithm is not polynomial.…”

Section: Applicationsmentioning

confidence: 99%

Approximating the Existential Theory of the Reals

Deligkas

Fearnley

Melissourgos

et al. 2018

Web and Internet Economics

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The existential theory of the reals (ETR) consists of existentially quantified boolean formulas over equalities and inequalities of real-valued polynomials. We propose the approximate existential theory of the reals (ǫ-ETR), in which the constraints only need to be satisfied approximately. We first show that unconstrained ǫ-ETR = ETR, and then study the ǫ-ETR problem when the solution is constrained to lie in a given convex set. Our main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games. It states that if an ETR problem has an exact solution, then it has a k-uniform approximate solution, where k depends on various properties of the formula. A consequence of our theorem is that we obtain a quasi-polynomial time approximation scheme (QPTAS) for a fragment of constrained ǫ-ETR. We use our theorem to create several new PTAS and QPTAS algorithms for problems from a variety of fields.

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Exact algorithms for solving stochastic games

Cited by 43 publications

References 14 publications

Bounds for the Condition Number of Polynomials Systems with Integer Coefficients

Bounds for the Condition Number of Polynomials Systems with Integer Coefficients

Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

Approximating the Existential Theory of the Reals

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