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63
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 190 (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
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 37 (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
FIR Filter Design via Spectral Factorization and Convex Optimization
, 1997
"... We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the filter coefficients as the variables and the frequency response bounds as constraints, are in general nonconvex. Usin ..."
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Cited by 35 (6 self)
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We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the filter coefficients as the variables and the frequency response bounds as constraints, are in general nonconvex. Using a change of variables and spectral factorization, we can pose such problems as linear or nonlinear convex optimization problems. As a result we can solve them efficiently (and globally) by recently developed interiorpoint methods. We describe applications to filter and equalizer design, and the related problem of antenna array weight design.
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...
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 th ..."
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Cited by 18 (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...
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) f ..."
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Cited by 18 (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 16 (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).
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.