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131
Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time
, 2003
"... We introduce the smoothed analysis of algorithms, which continuously interpolates between the worstcase and averagecase analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We me ..."
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Cited by 205 (14 self)
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We introduce the smoothed analysis of algorithms, which continuously interpolates between the worstcase and averagecase analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has smoothed complexity polynomial in the input size and the standard deviation of
SCIP: solving constraint integer programs
, 2009
"... Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), wh ..."
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Cited by 116 (0 self)
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Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and noncommercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly nonlinear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current stateoftheart techniques for proving the validity of properties on circuits containing arithmetic.
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 62 (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...
Receding Horizon Control of Nonlinear Systems: A Control . . .
, 2000
"... n Automatic Control, pages 898 907, 1990. J. Shamma and M. Athans. Guaranteed properties of gain scheduled control for linear parametervarying plants. Automatica, pages 559 564, 1991. J. Shamma and M. Athans. Gainscheduling: Potential hazards and possible remedies. IEEE Control Systems Magazine, ..."
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Cited by 61 (5 self)
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n Automatic Control, pages 898 907, 1990. J. Shamma and M. Athans. Guaranteed properties of gain scheduled control for linear parametervarying plants. Automatica, pages 559 564, 1991. J. Shamma and M. Athans. Gainscheduling: Potential hazards and possible remedies. IEEE Control Systems Magazine, 12(3):101 107, June 1992. [Sch96] A. Schwartz. Theory and Implementation of Numerical Methods Based on RungeKutta Integration for Optimal Control Problems. PhD Disser tation, University of California, Berkeley, 1996. [SCH+00] M. Sznaier, J. Cloutier, R. Hull, D. Jacques, and C. Mracek. Reced ing horizon control lyapunov function approach to suboptimal regula tion of nonlinear systems. Journal of Guidance, Control, and Dynamics, 23(3):399 405, 2000. [SD90] M. Sznaier and M. J. Damborg. Heuristically enhanced feedback con trol of constrained discretetime linear systems. Automatica, 26:521 532, 1990. [SMR99] P. Scokaert, D. Mayne, and J. Rawlings. Suboptimal model predictive cont
Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2012
"... We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."
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Cited by 54 (13 self)
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We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
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 ..."
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Cited by 48 (4 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.
A counterexample to the Hirsch conjecture
, 2011
"... The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, that any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the ..."
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Cited by 42 (2 self)
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The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, that any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5dimensional polytope with 48 facets which violates a certain generalization of the dstep conjecture of Klee and Walkup.
Real Quantifier Elimination in Practice
 Algorithmic Algebra and Number Theory
, 1998
"... We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applicat ..."
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Cited by 41 (6 self)
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We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applications in various fields of science, engineering, and economics.