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46
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 & Co ..."
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Cited by 188 (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
Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
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Cited by 65 (14 self)
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We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
Exploiting multiantennas for opportunistic spectrum sharing in cognitive radio networks
 IEEE J. Select. Topics in Signal Processing
, 2008
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ConditionBased Complexity Of Convex Optimization In Conic Linear Form Via The Ellipsoid Algorithm
, 1998
"... A convex optimization problem in conic linear form is an optimization problem of the form CP (d) : maximize c T ..."
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Cited by 40 (17 self)
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A convex optimization problem in conic linear form is an optimization problem of the form CP (d) : maximize c T
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...
On the Enumeration of Inscribable Graphs
 Manuscript, NEC Research Institute
, 1991
"... We explore the question of counting, and estimating the number and the fraction of, inscribable graphs. In particular we will concern ourselves with the number of inscribable and circumscribable maximal planar graphs (synonym: simplicial polyhedra) on V vertices, or, dually, the number of circumscr ..."
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Cited by 21 (4 self)
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We explore the question of counting, and estimating the number and the fraction of, inscribable graphs. In particular we will concern ourselves with the number of inscribable and circumscribable maximal planar graphs (synonym: simplicial polyhedra) on V vertices, or, dually, the number of circumscribable and inscribable trivalent (synonyms: 3regular, simple) polyhedra. For small V we provide computergenerated tables. Asymptotically for large V we will prove bounds showing that these graphs are exponentially numerous, but, viewed as a fraction of all maximal planar graphs, they are exponentially rare. Many of our results are based on a lemma, the "strong 01 law for maximal planar graphs," of independent interest. This is part of a series of TMs exploring graphtheoretic consequences of the recent RivinSmith characterization of "inscribable graphs." (A graph is a set of "vertices," some pairs of which are joined by "edges." A graph is "inscribable" if it is the 1skeleton of a conve...
A New Condition Measure, PreConditioners, and Relations between Different Measures of Conditioning for Conic Linear Systems
, 2001
"... In recent years, a body of research into "condition numbers" for convex optimization has been developed, aimed at capturing the intuitive notion of problem behavior. This research has been shown to be relevant in studying the efficiency of algorithms (including interiorpoint algorithms) for convex ..."
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Cited by 19 (6 self)
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In recent years, a body of research into "condition numbers" for convex optimization has been developed, aimed at capturing the intuitive notion of problem behavior. This research has been shown to be relevant in studying the efficiency of algorithms (including interiorpoint algorithms) for convex optimization as well as other behavioral characteristics of these problems such as problem geometry, deformation under data perturbation, etc. This paper studies measures of conditioning for a conic linear system of the form (FP d ): Ax = b; x 2 CX , whose data is d = (A; b). We present a new measure of conditioning, denoted d , and we show implications of d for problem geometry and algorithm complexity, and demonstrate that the value of = d is independent of the speci c data representation of (FP d ). We then prove certain relations among a variety of condition measures for (FP d ), including d , d , d , and C(d). We discuss some drawbacks of using the condition number C(d) as the sole measure of conditioning of a conic linear system, and we introduce the notion of a "preconditioner" for (FP d ) which results in an equivalent formulation (FP ~ d ) of (FP d ) with a better condition number C( ~ d). We characterize the best such preconditioner and provide an algorithm and complexity analysis for constructing an equivalent data instance ~ d whose condition number C( ~ d) is within a known factor of the best possible.
Condition Number Complexity of an Elementary Algorithm for Resolving a Conic Linear System
, 1997
"... We develop an algorithm for resolving a conic linear system (FP d ), which is a system of the form (FP d ): b Ax 2 C Y x 2 CX ; where CX and C Y are closed convex cones, and the data for the system is d = (A; b). ..."
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Cited by 17 (4 self)
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We develop an algorithm for resolving a conic linear system (FP d ), which is a system of the form (FP d ): b Ax 2 C Y x 2 CX ; where CX and C Y are closed convex cones, and the data for the system is d = (A; b).
Towards a Practical Volumetric Cutting Plane Method for Convex Programming
 SIAM Journal on Optimization
, 1997
"... We consider the volumetric cutting plane method for finding a point in a convex set C ae ! n that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed directly through the current point, and show that this "central cut" version of the method c ..."
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Cited by 16 (2 self)
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We consider the volumetric cutting plane method for finding a point in a convex set C ae ! n that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed directly through the current point, and show that this "central cut" version of the method can be implemented using no more than 25n constraints at any time. Currently visiting the Center for Operations Research and Econometrics, Catholic University of Louvain, LouvainlaNeuve, Belgium, with support from a CORE fellowship. 1 Introduction Let C ae ! n be a convex set. Given a point ¯ x 2 ! n , a separation oracle for C either reports that ¯ x 2 C, or returns a separating hyperplane a 2 ! n such that a T x ? a T ¯ x for every x 2 C. The convex feasibility problem is to use such an oracle to find a point in C, or prove that the volume of C must be less than that of an ndimensional sphere of radius 2 \GammaL , for given L ? 0. It is well known [9] that a variety of...
Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 16 (6 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.