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26
The nonlinear geometry of linear programming I. Affine and Projective Scaling Techniques
, 1986
"... 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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Cited by 83 (0 self)
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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
An update on the Hirsch conjecture,
 Jahresber. Dtsch. Math.Ver.
, 2010
"... Abstract The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample t ..."
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Abstract The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5dimensional polytope with 48 facets that violates a certain generalization of the dstep conjecture of Klee and Walkup.
Precise relational invariants through strategy iteration
 In CSL
, 2007
"... Abstract. We present a practical algorithm for computing exact least solutions of systems of equations over the rationals with addition, multiplication with positive constants, minimum and maximum. The algorithm is based on strategy improvement combined with solving linear programming problems for ..."
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Cited by 28 (8 self)
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Abstract. We present a practical algorithm for computing exact least solutions of systems of equations over the rationals with addition, multiplication with positive constants, minimum and maximum. The algorithm is based on strategy improvement combined with solving linear programming problems for each selected strategy. We apply our technique to compute the abstract least fixpoint semantics of affine programs over the relational template constraint matrix domain [20]. In particular, we thus obtain practical algorithms for computing the abstract least fixpoint semantics over the zone and octagon abstract domain. 1
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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Cited by 20 (2 self)
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In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1
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
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
Improving Strategies via SMT Solving
, 2011
"... We consider the problem of computing numerical invariants of programs by abstract interpretation. Our method eschews two traditional sources of imprecision: (i) the use of widening operators for enforcing convergence within a finite number of iterations (ii) the use of merge operations (often, conve ..."
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Cited by 10 (6 self)
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We consider the problem of computing numerical invariants of programs by abstract interpretation. Our method eschews two traditional sources of imprecision: (i) the use of widening operators for enforcing convergence within a finite number of iterations (ii) the use of merge operations (often, convex hulls) at the merge points of the control flow graph. It instead computes the least inductive invariant expressible in the domain at a restricted set of program points, and analyzes the rest of the code en bloc. We emphasize that we compute this inductive invariant precisely. For that we extend the strategy improvement algorithm of Gawlitza and Seidl [17]. If we applied their method directly, we would have to solve an exponentially sized system of abstract semantic equations, resulting in memory exhaustion. Instead, we keep the system implicit and discover strategy improvements using SAT modulo real linear arithmetic (SMT). For evaluating strategies we use linear programming. Our algorithm has low polynomial space complexity and performs for contrived examples in the worst case exponentially many strategy improvement steps; this is unsurprising, since we show that the associated abstract reachability problem is Π p 2complete.
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