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23
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 189 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
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 145 (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
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
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Cited by 69 (6 self)
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CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the
Smoothed Analysis of Termination of Linear Programming Algorithms
"... We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng ..."
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Cited by 21 (3 self)
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We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng
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 shadowvertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an ar ..."
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Cited by 18 (3 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 shadowvertex 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. SpielmanTeng’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 phaseI for solving arbitrary linear programs, which is of independent interest. Instead of finding a vertex of a feasible set, we add a vertex at
A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. Th ..."
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Cited by 9 (1 self)
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The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...
On Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems
, 2003
"... In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of ..."
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Cited by 8 (5 self)
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In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of the distribution tails, we improve the existing moment estimates, and we prove various limit theorems for the cases where either n or m and n tend to in nity. Our results are of complexity theoretic interest, because interiorpoint methods and relaxation methods for the solution of Ax 0, x 6= 0 have running times that are bounded in terms of log C (A) and C (A) respectively. AMS Classi cation: primary 90C31,15A52; secondary 90C05,90C60,62H10. Key Words: condition number, random matrices, linear programming, probabilistic analysis, complexity theory.
ON THE EXPECTED NUMBER OF LINEAR COMPLEMENTARITY CONES INTERSECTED BY RANDOM AND SEMIRANDOM RAYS
, 1986
"... Lemke's algorithm for the linear complementarity problem follows a ray which leads from a certain fixed point (traditionally, the point (1,..., I)~) to the point given in the problem. The problem also induces a set of 2 " cones, and a question which is relevant to the probabilistic ana ..."
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Cited by 4 (1 self)
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Lemke's algorithm for the linear complementarity problem follows a ray which leads from a certain fixed point (traditionally, the point (1,..., I)~) to the point given in the problem. The problem also induces a set of 2 &quot; cones, and a question which is relevant to the probabilistic analysis of Lemke's algorithm is to estimate the expected number of times a (semirandom) ray intersects the boundary between two adjacent cones. When the problem is sampled from a spherically symmetric distribution this number turns out to be exponential. For an ndimensional problem the natural logarithm of this number is equal to ln(r)n + o(n), where T is approximately 1.151222. This number stands in sharp contrast with the expected number of cones intersected by a ray which is determined by two random points (call it random). The latter is only (n/2)+ 1. The discrepancy between linear behavior (under the 'random ' assumption) and exponential behavior (under the 'semirandom ' assumption) has implications with respect to recent analyses of the average complexity of the linear programming problem. Surprisingly, the semirandom case is very sensitive to the fixed point of the ray, even when that point is confined to the positive orthant. We show that for points of the form (E, E',..., E &quot;) ~ the expected number of facets of cones cut by a semirandom ray tends to in2+2n when E tends to zero.