A Combinatorial Strongly Subexponential Strategy Improvement Algorithm for Mean Payoff Games

Björklund, Henrik; Sandberg, Sven; Vorobyov, Sergei

doi:10.1007/978-3-540-28629-5_52

Cited by 50 publications

(87 citation statements)

References 20 publications

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“…Independently of our results, Björklund et al [BSV2], [BSV3] developed ad hoc strongly e O( √ n log n) algorithms for PGs and for the decision problem corresponding to MPGs (they also provided a log W · e O( √ n log n) algorithm for MPGs). We summarize the previous best results and our results in Table 1.…”

Section: Introduction Sharir and Welzlmentioning

confidence: 88%

“…We also give alternative simple proofs for the best known upper bounds for PGs and binary SSGs. While Ludwig [L] and Björkland et al [BSV1]- [BSV3] developed ad hoc algorithms for each of the specific games they solved, we use only one unifying algorithm for all of these five games-the LP-type algorithm of Sharis and Welzl [SW].…”

Section: Introduction Sharir and Welzlmentioning

confidence: 99%

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Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All LP-Type Problems

Halman

2007

Algorithmica

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We show that a Simple Stochastic Game (SSG) can be formulated as an LP-type problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LP-type problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known for binary SSGs [L]). Using known reductions between various games, we achieve the first strongly subexponential solutions for Discounted and Mean Payoff Games. We also give alternative simple proofs for the best known upper bounds for Parity Games and binary SSGs.To the best of our knowledge, the LP-type framework has been used so far only in order to yield linear or close to linear time algorithms for various problems in computational geometry and location theory. Our approach demonstrates the applicability of the LP-type framework in other fields, and for achieving subexponential algorithms. Introduction. Sharir and Welzl [SW] defined a model which generalizesLinear Programming (LP) and called it the LP-type model (see definitions in Section 2.2). An LP-type problem of combinatorial dimension d, where d is independent of the size n of the problem, is called fixed dimensional. Several algorithms that solve LP-type problems in time linear in n are known, such as the ones of Sharir and Welzl [SW] or Kalai [Ka]. The O(n) time algorithm of Clarkson [Cl], which was originally formulated to solve LP, fits the LP-type model as well [CM], [GW1]. By formulating problems as fixeddimensional LP-type problems, and using the LP-type algorithms, one can obtain linear time algorithms to various optimization problems, mainly in computational geometry and location theory, as shown in [A] and [MSW].The algorithms of [Ka] and [SW] run in time subexponential in d. In this paper we use the LP-type framework in order to give the first strongly subexponential solution for Simple Stochastic Games, Discounted Payoff Games and Mean Payoff Games (defined below). To the best of our knowledge, this is the first application of the LP-type framework for solving a problem which is neither in computational geometry nor in location theory. Moreover, it is the first application of variable-dimensional LP-type problems.A Simple Stochastic Game (SSG) is defined on a directed graph with three types of vertices, min, max and average, along with two sink vertices, the 0-sink and the 1-sink.

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Section: Introduction Sharir and Welzlmentioning

confidence: 88%

Section: Introduction Sharir and Welzlmentioning

confidence: 99%

Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All LP-Type Problems

Halman

2007

Algorithmica

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“…For this, it chooses a random hypervertex J I in G I (more precisely, the one where J i = j ) and recursively evaluates the sink K I among the remaining | i | − 1 hypervertices. 3 If K I is not the sink w.r.t. ψ I yet, one more evaluation of the true sink (K j) I = J I is necessary.…”

Section: Its Solution Ismentioning

confidence: 99%

“…By the reductions of Björklund et al [2,3] (and the earlier reduction of Ludwig for binary SSG [34]), we can interpret the Hoffman-Karp algorithm as a sink-finding method for grid AUSO. In the USO world, this method is known as the bottomantipodal algorithm, and there are exponential lower bounds for its performance [40].…”

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confidence: 98%

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Unique Sink Orientations of Grids

2007

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We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games.We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of blocks.

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Unique Sink Orientations of Grids

Gärtner

Morris

Rüst

2005

Integer Programming and Combinatorial Optimization

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We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices and generalized linear complementarity problems over P-matrices (PGLCP). We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of blocks.

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A Combinatorial Strongly Subexponential Strategy Improvement Algorithm for Mean Payoff Games

Cited by 50 publications

References 20 publications

Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All LP-Type Problems

Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All LP-Type Problems

Unique Sink Orientations of Grids

Unique Sink Orientations of Grids

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