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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 worst-case and average-case 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 202 (12 self)
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We introduce the smoothed analysis of algorithms, which continuously interpolates between the worst-case and average-case 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
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 Lemke-Howson 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 PPA ..."
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Cited by 47 (1 self)
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The Lemke-Howson 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 Lemke--Howson 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 d-space.
An update on the Hirsch conjecture,
- Jahresber. Dtsch. Math.-Ver.
, 2010
"... Abstract The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample t ..."
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Cited by 41 (3 self)
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Abstract The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, 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 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
Linear Programming, the Simplex Algorithm and Simple Polytopes
- Math. Programming
, 1997
"... In the first part of the paper we survey some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot ru ..."
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Cited by 31 (1 self)
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In the first part of the paper we survey some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a d-dimensional polyhedron (briefly, a d-polyhedron) if the points in P affinely span R d . A convex d-dimensional polytope (briefly, a d-polytope) is a bounded convex d-polyhedron. Alternatively, a convex d-polytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a d-polyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a k-face of P . The empty set and P itself are...
A randomized polynomial-time simplex algorithm for linear programming
- IN STOC
, 2006
"... We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. We begin by reducing the input linear program to ..."
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Cited by 28 (5 self)
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We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. We begin by reducing the input linear program to a special form in which we merely need to certify boundedness. As boundedness does not depend upon the right-hand-side vector, we run the shadow-vertex simplex method with a random right-hand-side vector. Thus, we do not need to bound the diameter of the original polytope. Our analysis rests on a geometric statement of independent interest: given a polytope Ax ≤ b in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial.
A survey of linear programming in randomized subexponential time
- SIGACT News
, 1995
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Beyond Hirsch conjecture: Walks on random polytopes and smoothed complexity of the simplex method
- In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
, 2006
"... Abstract. The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow-vertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an ar ..."
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Cited by 24 (2 self)
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Abstract. The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow-vertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/σ. We show that the length of walk in the simplex method is actually polylogarithmic in the number of constraints n. Spielman-Teng’s bound on the walk was O ∗ (n 86 d 55 σ −30), up to logarithmic factors. We improve this to O(log 7 n(d 9 + d 3 σ −4)). This shows that the tight Hirsch conjecture n − d on the length of walk on polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. We propose a randomized phase-I for solving arbitrary linear programs, which is of independent interest. Instead of finding a vertex of a feasible set, we add a vertex at
