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17
A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games
 DISCRETE APPLIED MATHEMATICS
, 2004
"... We suggest the first strongly subexponential and purely combinatorial algorithm for solving the mean payoff games problem. It is based on iteratively improving the longest shortest distances to a sink in a possibly cyclic directed graph. We identify a new “controlled” version of the shortest paths p ..."
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Cited by 40 (4 self)
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We suggest the first strongly subexponential and purely combinatorial algorithm for solving the mean payoff games problem. It is based on iteratively improving the longest shortest distances to a sink in a possibly cyclic directed graph. We identify a new “controlled” version of the shortest paths problem. By selecting exactly one outgoing edge in each of the controlled vertices we want to make the shortest distances from all vertices to the unique sink as long as possible. Under reasonable assumptions the problem belongs to the complexity class NP∩coNP. Mean payoff games are easily reducible to this problem. We suggest an algorithm for computing longest shortest paths. Player Max selects a strategy (one edge in each controlled vertex) and player Min responds by evaluating shortest paths to the sink in the remaining graph. Then Max locally changes choices in controlled vertices looking at attractive switches that seem to increase shortest paths lengths (under the current evaluation). We show that this is a monotonic strategy improvement, and every locally optimal strategy is globally optimal. This allows us to construct a randomized algorithm of complexity min(poly · W, 2 O( √ n log n)), which is simultaneously pseudopolynomial (W is the maximal absolute edge weight) and subexponential in the number of vertices n. All previous algorithms for mean payoff games were either exponential or pseudopolynomial (which is purely exponential for exponentially large edge weights).
A discrete subexponential algorithm for parity games
 STACS’03
, 2003
"... We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly ..."
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Cited by 34 (8 self)
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We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly
Convex hull realizations of the multiplihedra
, 2007
"... Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents ..."
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Cited by 13 (4 self)
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Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
"... ..."
Combinatorial linear programming: Geometry can help
 Proc. 2nd Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), Lecture Notes in Computer Science 1518
, 1998
"... We consider a class A of generalized linear programs on the dcube (due to Matousek) and prove that Kalai's subexponential simplex algorithm RandomFacet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general inst ..."
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Cited by 9 (2 self)
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We consider a class A of generalized linear programs on the dcube (due to Matousek) and prove that Kalai's subexponential simplex algorithm RandomFacet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general instances in A. Thus, we identify a "geometric" property of linear programming that goes beyond all abstract notions previously employed in generalized linear programming frameworks, and that can be exploited by the simplex method in a nontrivial setting.
Unique sink orientations of grids
 Proc. 11th Conference on Integer Programming and Combinatorial Optimization (IPCO
, 2005
"... We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of su ..."
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Cited by 7 (4 self)
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We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (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 HoltKlee condition known to hold for polytope digraphs, and we give the first expected lineartime algorithms for solving PGLCP with a fixed number of blocks.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Randomized Subexponential Algorithms for Infinite Games
, 2004
"... The complexity of solving infinite games, including parity, mean payoff, and simple stochastic games, is an important open problem in verification, automata theory, and complexity theory. In this paper we develop an abstract setting for studying and solving such games, as well as related problems, b ..."
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Cited by 6 (0 self)
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The complexity of solving infinite games, including parity, mean payoff, and simple stochastic games, is an important open problem in verification, automata theory, and complexity theory. In this paper we develop an abstract setting for studying and solving such games, as well as related problems, based on function optimization over certain discrete structures. We introduce new classes of completely localglobal (CLG) and recursively localglobal (RLG) functions, and show that strategy evaluation functions for parity and simple stochastic games belong to these classes. We also establish a relation to the previously wellstudied completely unimodal (CU) and localglobal functions. A number of nice properties of CLGfunctions are proved. In this setting, we survey several randomized optimization algorithms appropriate for CU, CLG, and RLGfunctions. We show that the subexponential algorithms for linear programming by Kalai and Matouˇsek, Sharir, and Welzl, can be adapted to optimizing the functions we study, with preserved subexponential expected running time. We examine the relations to two other abstract frameworks for subexponential
Two new bounds for the randomedge simplex algorithm, preprint, arXiv: math.CO/0502025
, 2005
"... Abstract. We prove that the RandomEdge simplex algorithm requires an expected number of at most 13n / √ d pivot steps on any simple dpolytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bo ..."
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Cited by 2 (0 self)
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Abstract. We prove that the RandomEdge simplex algorithm requires an expected number of at most 13n / √ d pivot steps on any simple dpolytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial dcubes, the trivial upper bound of 2 d on the performance of RandomEdge can asymptotically be improved by any desired polynomial factor in d. 1.