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45
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 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
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 38 (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 26 (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.
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...
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 15 (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.
On peak versus average interference power constraints for protecting primary users in cognitive radio networks
 IEEE Trans. Wireless Commun
, 2009
"... This paper considers spectrum sharing for wireless communication between a cognitive radio (CR) link and a primary radio (PR) link. It is assumed that the CR protects the PR transmission by applying the socalled interferencetemperature constraint, whereby the CR is allowed to transmit regardless o ..."
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Cited by 13 (6 self)
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This paper considers spectrum sharing for wireless communication between a cognitive radio (CR) link and a primary radio (PR) link. It is assumed that the CR protects the PR transmission by applying the socalled interferencetemperature constraint, whereby the CR is allowed to transmit regardless of the PR’s on/off status provided that the resultant interference power level at the PR receiver is kept below some predefined threshold. For the fading PR and CR channels, the interferencepower constraint at the PR receiver is usually one of the following two types: One is to regulate the average interference power (AIP) over all the fading states, while the other is to limit the peak interference power (PIP) at each fading state. From the CR’s perspective, given the same average and peak power threshold, the AIP constraint is more favorable than the PIP counterpart because of its more flexibility for dynamically allocating transmit powers over the fading states. On the contrary, from the perspective of protecting the PR, the more restrictive PIP constraint appears at a first glance to be a better option than the AIP. Some surprisingly, this paper shows that in terms of various forms of capacity limits achievable for the PR fading channel, e.g., the ergodic and outage capacities, the AIP constraint is also superior over the PIP. This result is based upon an interesting interference diversity phenomenon, i.e., randomized interference powers over the fading states in the AIP case are more advantageous over deterministic ones in the PIP case for minimizing the resultant PR capacity losses. Therefore, the AIP constraint results in larger fading channel capacities than the PIP for both the CR and PR transmissions. Index Terms Cognitive radio, spectrum sharing, interference temperature, interference diversity, fading channel capacity.