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The nonlinear geometry of linear programming IV. Hilbert geometry, in preparation
"... This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope ..."
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This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure we study is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. In order to study Ptrajectories we also study a related vector field on the linear programming polytope, which we call the affine scaling vector field, and its associated trajectories, called Atrajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. These affine and projective scaling vector fields are each defined for liner programs of a special form, called strict standard form and canonical form, respectively. This paper defines and presents basic properties of Ptrajectories and Atrajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for Ptrajectories and Atrajectories. It presents Karmarkar’s interpretation of Atrajectories as steepest descent paths of the objective function 〈c, x 〉 with respect to the Riemannian _ dx
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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. 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...
Games Computers Play: GameTheoretic Aspects of Computing
 In
, 1992
"... this article is on protocols allowing the wellfunctioning parts of such a large and complex system to carry out their work despite the failure of others. Many deep and interesting results on such problems have been discovered by computer scientists in recent years, the incorporation of which into g ..."
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this article is on protocols allowing the wellfunctioning parts of such a large and complex system to carry out their work despite the failure of others. Many deep and interesting results on such problems have been discovered by computer scientists in recent years, the incorporation of which into game theory can greatly enrich this field
Sublineartime algorithms
 In Oded Goldreich, editor, Property Testing, volume 6390 of Lecture Notes in Computer Science
, 2010
"... In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1 ..."
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In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1
Lexicographic Maxmin Fairness for Data Collection in Wireless Sensor Networks
"... Abstract—The ad hoc deployment of a sensor network causes unpredictable patterns of connectivity and varied node density, resulting in uneven bandwidth provisioning on the forwarding paths. When congestion happens, some sensors may have to reduce their data rates. It is an interesting but difficult ..."
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Abstract—The ad hoc deployment of a sensor network causes unpredictable patterns of connectivity and varied node density, resulting in uneven bandwidth provisioning on the forwarding paths. When congestion happens, some sensors may have to reduce their data rates. It is an interesting but difficult problem to determine which sensors must reduce rates and how much they should reduce. This paper attempts to answer a fundamental question about congestion resolution: What are the maximum rates at which the individual sensors can produce data without causing congestion in the network and unfairness among the peers? We define the maxmin optimal rate assignment problem in a sensor network, where all possible forwarding paths are considered. We provide an iterative linear programming solution, which finds the maxmin optimal rate assignment and a forwarding schedule that implements the assignment in a lowrate sensor network. We prove that there is one and only one such assignment for a given configuration of the sensor network. We also study the variants of the maxmin fairness problem in sensor networks. Index Terms—Multipath maxmin fairness, wireless sensor networks, data collection applications, iterative linear programming. 1
A deterministic polynomialtime approximation scheme for counting knapsack solutions
, 1008
"... Abstract. Given n elements with nonnegative integer weights w1,...,wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates ..."
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Abstract. Given n elements with nonnegative integer weights w1,...,wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error 1±ε in time polynomial in n and 1/ε (fully polynomial approximation scheme). More precisely, our algorithm takes time O(n3ε−1 log(n/ε)). Our algorithm is based on dynamic programming. Previously, randomized polynomialtime approximation schemes were known first by Morris and Sinclair via Markov chain Monte Carlo techniques and subsequently by Dyer via dynamic programming and rejection sampling. Key words. approximate counting, knapsack, dynamic programming 1. Introduction. Randomized
Embedding Rivers in Triangulated Irregular Networks with Linear Programming
"... Data conflation is a major issue in GIS: different geospatial data sets covering overlapping regions, possibly obtained from different sources and using different acquisition techniques, need to be combined into one single consistent data set before the data can be analyzed. The most common occurren ..."
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Data conflation is a major issue in GIS: different geospatial data sets covering overlapping regions, possibly obtained from different sources and using different acquisition techniques, need to be combined into one single consistent data set before the data can be analyzed. The most common occurrence for hydrological applications is conflation of a digital elevation model and rivers. We assume that a triangulated irregular network (TIN) is given, and a subset of its edges are designated as river edges, each with a flow direction. The goal is to obtain a terrain where the rivers flow along valley edges, in the specified direction, while preserving the original terrain as much as possible. We study the problem of changing the elevations of the vertices to ensure that all the river edges become valley edges, while minimizing the total elevation change. We show that this problem can be solved using linear programming. However, several types of artifacts can occur in an optimal solution. We analyze which other criteria, relevant for hydrological applications, can be captured by linear constraints as well, in order to eliminate such artifacts. We implemented and tested the approach on real terrain and river data, and describe the results obtained with different variants of the algorithm. 1
Algorithms for LeximinOptimal Fair Policies in Repeated
"... Solutions to noncooperative multiagent systems often require achieving a joint policy which is as fair to all parties as possible. There are a variety of methods for determining the fairest such joint policy. One approach, min fairness, finds the policy which maximizes the minimum average reward gi ..."
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Solutions to noncooperative multiagent systems often require achieving a joint policy which is as fair to all parties as possible. There are a variety of methods for determining the fairest such joint policy. One approach, min fairness, finds the policy which maximizes the minimum average reward given to any agent. We focus on an extension, leximin fairness, which breaks ties among candidate policies by choosing the one which maximizes the secondtominimum average reward, then the thirdtominimum average reward, and so on. This method has a number of advantages over others in the literature, but has so far been littleused because of the computational cost in employing it to find the fairest policy. In this paper we propose a linear programming based algorithm for computing leximin fairness in repeated games which has a polynomial time complexity given certain reasonable assumptions. 1
OPTIMA Mathematical Optimization Society Newsletter
, 2011
"... with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meet ..."
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with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meetings, we are now heading towards the high point of 2012: the ISMP in Berlin. I hear from good sources that preparations are progressing well, and that all augurs are favourable. As you all know, several prizes will be awarded at the ISMP opening ceremony, recognizing the contributions or both younger and more senior colleagues. You undoubtedly have seen the various calls for nominations for the Dantzig, Lagrange, Fulkerson, BealeOrchardHays and Tucker prizes as well as that for the Paul Tseng lectureship. I encourage you to seriously consider nominating one or more optimization researchers for these prizes. These awards and the high scientific standards of their recipients not only recognize