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45
Hellytype theorems and generalized linear programming
 DISCRETE COMPUT. GEOM
, 1994
"... This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use the ..."
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Cited by 60 (0 self)
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This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...
A quasipolynomial bound for the diameter of graphs of polyhedra
 Bulletin Amer. Math. Soc
, 1992
"... Abstract. The diameter of the graph of a ddimensional polyhedron with n facets is at most nlog d+2 graph whose vertices are the extreme points of P and two vertices u and v are adjacent if the interval [v, u] is an extreme edge ( = 1dimensional face) of P. The diameter of the graph of P is denoted ..."
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Cited by 52 (4 self)
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Abstract. The diameter of the graph of a ddimensional polyhedron with n facets is at most nlog d+2 graph whose vertices are the extreme points of P and two vertices u and v are adjacent if the interval [v, u] is an extreme edge ( = 1dimensional face) of P. The diameter of the graph of P is denoted by δ(P). Let ∆(d, n) be the maximal diameter of the graphs of ddimensional polyhedra P with n facets. (A facet is a (d − 1)dimensional face.) Thus, P is the set of solutions of n linear inequalities in d variables. It is an old standing problem to determine the behavior of the function ∆(d, n). The value of ∆(d, n) is a lower bound for the number of iterations needed for Dantzig’s simplex algorithm for linear programming with any pivot rule. In 1957 Hirsch conjectured [2] that ∆(d, n) ≤ n−d. Klee and Walkup [6] showed that the Hirsch conjecture is false for unbounded polyhedra. They proved that for n ≥ 2d, ∆(d, n) ≥ n − d + [d/5]. This is the best known lower bound for ∆(d, n). The statement of the Hirsch conjecture for bounded polyhedra is still open. For a recent survey on the Hirsch conjecture and its relatives, see [5]. In 1967 Barnette proved [1, 3] that ∆(d, n) ≤ n3 d−3. An improved upper bound, ∆(d, n) ≤ n2 d−3, was proved in 1970 by Larman [7]. Barnette’s and Larman’s bounds are linear in n but exponential in the dimension √ d. In 1990 the first author [4] proved a subexponential bound ∆(d, n) ≤ 2 (n−d)log(n−d). The purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for ∆(d, n). Such a bound was proved by the first author in March 1991. The proof presented here is a substantial simplification that was subsequently found by the second author. See [4] for the original proof and related results. The existence of a polynomial (or even linear) upper bound for ∆(d, n) is still open. Recently, the first author found a randomized pivot rule for linear programming which requires an expected n 4 √ d (or less) arithmetic operations for every linear programming problem with d variables and n constraints.
A subexponential algorithm for abstract optimization problems
 SIAM J. Comput
, 1995
"... An Abstract Optimization Problem (AOP) is a triple (H, <, Φ) where H is a finite set, < a total order on 2 H and Φ an oracle that, for given F ⊆ G ⊆ H, either reports that F = min<{F ′  F ′ ⊆ G} or returns a set F ′ ⊆ G with F ′ < F. To solve the problem means to find the minimum set in H. We pr ..."
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Cited by 49 (5 self)
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An Abstract Optimization Problem (AOP) is a triple (H, <, Φ) where H is a finite set, < a total order on 2 H and Φ an oracle that, for given F ⊆ G ⊆ H, either reports that F = min<{F ′  F ′ ⊆ G} or returns a set F ′ ⊆ G with F ′ < F. To solve the problem means to find the minimum set in H. We present a randomized algorithm that solves any AOP with an expected number of at most e 2 √ n+O ( 4 √ n ln n) oracle calls, n = H. In contrast, any deterministic algorithm needs to make 2 n − 1 oracle calls in the worst case. The algorithm is applied to the problem of finding the distance between two nvertex (or nfacet) convex polyhedra in dspace, and to the computation of the smallest ball containing n points in dspace; for both problems we give the first subexponential bounds in the arithmetic model of computation.
Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game
 IN PROCEEDINGS OF THE ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2004
"... The LemkeHowson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive, elementary proof of existence of an equilibrium, by a typical "directed parity argument", which puts NASH into the complexity class PPAD. This pa ..."
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Cited by 46 (1 self)
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The LemkeHowson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive, elementary proof of existence of an equilibrium, by a typical "directed parity argument", which puts NASH into the complexity class PPAD. This paper presents a class of bimatrix games for which the LemkeHowson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring #((# 3=4 ) d ) many steps, where # is the Golden Ratio. The "parity argument" for NASH is thus explicitly shown to be inefficient. The games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in dspace.
The Many Facets of Linear Programming
, 2000
"... . We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction A ..."
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Cited by 25 (1 self)
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. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the fortyfifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twentyfifth of the awarding of the 1975 Nobe...
Hardtosolve bimatrix games
 ECONOMETRICA
, 2006
"... The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, th ..."
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Cited by 24 (1 self)
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The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in dspace. The construction is extended to nonsquare games where, in addition to exponentially long Lemke–Howson computations, finding an equilibrium by support enumeration takes on average exponential time.
Randomized Simplex Algorithms on KleeMinty Cubes
 COMBINATORICA
, 1994
"... We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes ..."
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Cited by 19 (6 self)
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We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the randomedge simplex algorithm on KleeMinty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a KleeMinty cube is exponential when all paths are taken with equal probability.
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
Decompositions of Simplicial Balls and Spheres With Knots Consisting of Few Edges
, 1999
"... Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] a ..."
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Cited by 16 (6 self)
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Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about nonshellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a nontrivial knot with "few edges" in a 3sphere or 3ball, and a similar hierarchy for 3balls with a knotted spanning arc that consists of "few edges."