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31
polymake: a Framework for Analyzing Convex Polytopes
, 1999
"... polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few ..."
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Cited by 168 (21 self)
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polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tu-berlin.de/diskregeom/polymake/.
Combinatorial structure and randomized subexponential algorithms for infinite games
, 2005
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Convex Combinatorial Optimization
, 2004
"... We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications. ..."
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Cited by 18 (7 self)
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We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.
A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games
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The random facet simplex algorithm on combinatorial cubes
- Random Structures & Algorithms
, 2001
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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 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 su ..."
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Cited by 11 (4 self)
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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.
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 d-cube (due to Matousek) and prove that Kalai's subexponential simplex algorithm Random-Facet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general ..."
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Cited by 10 (2 self)
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We consider a class A of generalized linear programs on the d-cube (due to Matousek) and prove that Kalai's subexponential simplex algorithm Random-Facet 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.
Diameter of Polyhedra: Limits of Abstraction
, 2009
"... We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that ..."
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Cited by 9 (2 self)
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We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear lower bound.