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31
Voronoi cells, probabilistic bounds and hypothesis testing in mixed integer linear models
 IEEE Trans. Inf. Theory
, 2006
"... (c)2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other ..."
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(c)2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound t ..."
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Cited by 7 (0 self)
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In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
One Line and n Points
 Proc. 33rd Ann. ACM Symp. on the Theory of Computing (STOC
, 2003
"... We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the RandomEdge simplex algorithm on simple polytopes with n facets in dimension n  2. We obtain a tight O(log² n) bound for the expected number of pivot steps. This is ..."
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Cited by 7 (2 self)
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We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the RandomEdge simplex algorithm on simple polytopes with n facets in dimension n  2. We obtain a tight O(log&sup2; n) bound for the expected number of pivot steps. This is the first nontrivial bound for RandomEdge which goes beyond bounds for specific polytopes. The process itself can be interpreted as a simple algorithm for certain 2variable linear programming problems, and we prove a tight &Theta;(n) bound for its expected runtime.
A subexponential lower bound for the Random Facet algorithm for Parity Games
"... Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of t ..."
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Cited by 6 (5 self)
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Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of turnbased Stochastic Mean Payoff Games. It is a major open problem whether these game families can be solved in polynomial time. The currently fastest algorithms for the solution of all these games are adaptations of the randomized generalizationof linear programming. We refer to the algorithm ofMatouˇsek, Sharir and Welzl as the Random Facet algorithm. The expected running time of these algorithmsis subexponential in the size of the game, i.e., 2
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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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).
The Number of UniqueSink Orientations of the Hypercube
"... Let Q d denote the graph of the ddimensional cube. A uniquesink orientation (USO) is an orientation of Q d such that every face of Qn has exactly one sink (vertex of outdegree 0); it does not have to be acyclic. USO have been studied as an abstract model for many geometric optimization problems ..."
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Let Q d denote the graph of the ddimensional cube. A uniquesink orientation (USO) is an orientation of Q d such that every face of Qn has exactly one sink (vertex of outdegree 0); it does not have to be acyclic. USO have been studied as an abstract model for many geometric optimization problems, such as linear programming, nding the smallest enclosing ball of a given point set, certain classes of convex programming, and certain linear complementarity problems. It is shown that the number of USO is d .
A Combinatorial Active Set Algorithm for Linear and Quadratic Programming, Under revision
, 2008
"... Abstract. We propose an algorithm for linear programming, which we call the Sequential Projection algorithm. This new approach is a primal improvement algorithm that keeps both a feasible point and an active set, which uniquely define an improving direction. Like the simplex method, the complexity o ..."
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Abstract. We propose an algorithm for linear programming, which we call the Sequential Projection algorithm. This new approach is a primal improvement algorithm that keeps both a feasible point and an active set, which uniquely define an improving direction. Like the simplex method, the complexity of this algorithm need not depend explicitly on the size of the numbers of the problem instance. Unlike the simplex method, however, our approach is not an edgefollowing algorithm, and the active set need not form a row basis of the constraint matrix. Moreover, the algorithm has a number of desirable properties that ensure that it is not susceptible to the simple pathological examples (e.g., the KleeMinty problems) that are known to cause the simplex method to perform an exponential number of iterations. We also show how to randomize the algorithm so that it runs in an expected time that is on the order of mn 2 log n for most LP instances. This bound is strongly subexponential in the size of the problem instance (i.e., it does not depend on the size of the data, and it can be bounded by a function that grows more slowly than 2 m, where m is the number of constraints in the problem). Moreover, to the best of our knowledge, this is the fastest known randomized algorithm for linear programming whose running time does not depend on the size of the numbers defining the problem instance. Many of our results generalize in a straightforward manner to mathematical programs that maximize a concave quadratic objective function over linear constraints (i.e., quadratic programs), and we discuss these extensions as well.
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
Errata for: A subexponential lower bound for the Random Facet algorithm for Parity Games
, 2014
"... In [Friedmann, Hansen, and Zwick (2011)] and we claimed that the expected number of pivoting steps performed by the RandomFacet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by RandomFacet∗, a variant of RandomFacet that bases i ..."
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In [Friedmann, Hansen, and Zwick (2011)] and we claimed that the expected number of pivoting steps performed by the RandomFacet algorithm of Kalai and of Matoušek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by RandomFacet∗, a variant of RandomFacet that bases its random decisions on one random permutation. We then obtained a lower bound on the expected number of pivoting steps performed by RandomFacet ∗ and claimed that the same lower bound holds also for RandomFacet. Unfortunately, the claim that the expected number of steps performed by RandomFacet and RandomFacet ∗ are the same is false. We provide here simple examples that show that the expected number of steps performed by the two algorithms is not the same. 1