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12
Proportionate progress: A notion of fairness in resource allocation
 Algorithmica
, 1996
"... Given a set of n tasks and m resources, where each task x has a rational weight x:w = x:e=x:p; 0 < x:w < 1, a periodic schedule is one that allocates a resource to a task x for exactly x:e time units in each interval [x:p k; x:p (k + 1)) for all k 2 N. We de ne a notion of proportionate progre ..."
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Cited by 322 (26 self)
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Given a set of n tasks and m resources, where each task x has a rational weight x:w = x:e=x:p; 0 < x:w < 1, a periodic schedule is one that allocates a resource to a task x for exactly x:e time units in each interval [x:p k; x:p (k + 1)) for all k 2 N. We de ne a notion of proportionate progress, called Pfairness, and use it to design an e cient algorithm which solves the periodic scheduling problem. Keywords: Euclid's algorithm, fairness, network ow, periodic scheduling, resource allocation.
Algorithmic Geometry of Numbers
 Annual Review of Comp. Sci
, 1987
"... this article  Algorithmic Geometry of Numbers. The fundamental basis reduction algorithm of Lov'asz which first appeared in Lenstra, Lenstra, Lov'asz [46] was used in Lenstra's algorithm for integer programming and has since been applied in myriad contextsstarting with factorization ..."
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Cited by 53 (0 self)
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this article  Algorithmic Geometry of Numbers. The fundamental basis reduction algorithm of Lov'asz which first appeared in Lenstra, Lenstra, Lov'asz [46] was used in Lenstra's algorithm for integer programming and has since been applied in myriad contextsstarting with factorization of polynomials (A.K. Lenstra, [45]). Classical Geometry of Numbers has a special feature in that it studies the geometric properties of (convex) sets like volume, width etc. which come from the realm of continuous mathematics in relation to lattices which are discrete objects. This makes it ideal for applications to integer programming and other discrete optimization problems which seem inherently harder than their "continuous" counterparts like linear programming. 1
NonStandard Approaches to Integer Programming
, 2000
"... In this survey we address three of the principle algebraic approaches to integer programming. After introducing... ..."
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Cited by 34 (4 self)
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In this survey we address three of the principle algebraic approaches to integer programming. After introducing...
Lifting 2integer knapsack inequalities
, 2003
"... In this paper we discuss the generation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The description of the basic polyhedra can be made in polynomial time. We us ..."
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Cited by 2 (0 self)
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In this paper we discuss the generation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The description of the basic polyhedra can be made in polynomial time. We use superadditive valid functions in order to obtain sequence independent lifting. 1
Cascading Knapsack Inequalities: Hidden Structure in some InventoryProductionDistribution problems
"... In the last decade MixedInteger Programming solvers have evolved enormously contributing to the widespread application of optimization in real world problems in industry. Nonetheless, it is paramount for practitioners to have basic knowledge on how these solvers work and to be able to identify mode ..."
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In the last decade MixedInteger Programming solvers have evolved enormously contributing to the widespread application of optimization in real world problems in industry. Nonetheless, it is paramount for practitioners to have basic knowledge on how these solvers work and to be able to identify model structures, so one can take full advantage of the machinery at hand. In this paper we present a reformulation to a simple problem that appears as subproblem in a vast majority of supply chain models, and we show the advantage of using suitable mathematical structures in the form of cascading knapsack inequalities to solve it. Moreover, we introduce new reformulations to some special cases, producing tighter linear relaxation and faster solution times. Key words: integer programming; knapsack inequalities; inventorydistribution planning; model reformulations History: 1.
Prtnted m U.S.A. INTEGER PROGRAMMING WITH A FIXED NUMBER OF VARIABLES*
"... It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers. The integer linear programming problem is formulated äs follows. Let n and m be positive integers, A an m X «matrix with integral ..."
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It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers. The integer linear programming problem is formulated äs follows. Let n and m be positive integers, A an m X «matrix with integral coefficients, and b e T&quot;. The question is to decide whether there exists a vector χ e / &quot; satisfying the System of m inequalities Ax < b. No algorithm for the solution of this problem is known which has a running time that is bounded by a polynomial function of the length of the data. This length may, for our purposes, be defined to be n · m · log(iz + 2), where a denotes the maximum of the absolute values of the coefficients of A and b. Indeed, no such polynomial algorithm is likely to exist, since the problem in question is NPcomplete [3], [12]. In this paper we consider the integer linear programming problem with a fixed value of n. In the case n = l it is trivial to design a polynomial algorithm for the solution of the problem. For n = 2, Hirschberg and Wong [5] and Kannan [6] have given polynomial algorithms in special cases. A complete treatment of the case n = 2 was
New Results on the Distance Between a Segment and Z2. Application to the Exact Rounding
"... This paper presents extensions to Lefèvre’s algorithm that computes a lower bound on the distance between a segment and a regular grid Z2. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or baseconver ..."
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This paper presents extensions to Lefèvre’s algorithm that computes a lower bound on the distance between a segment and a regular grid Z2. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or baseconversion functions. The proof that is presented here is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion. 1.
Abstract Theory and Methodology Analysis of upper bounds for the Pallet Loading Problem
, 1999
"... This paper is concerned with upper bounds for the wellknown Pallet Loading Problem �PLP), which is the problem of packing identical boxes into a rectangular pallet so as to maximize the number of boxes ®tted. After giving a comprehensive review of the known upper bounds in the literature, we conduc ..."
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This paper is concerned with upper bounds for the wellknown Pallet Loading Problem �PLP), which is the problem of packing identical boxes into a rectangular pallet so as to maximize the number of boxes ®tted. After giving a comprehensive review of the known upper bounds in the literature, we conduct a detailed analysis to determine which bounds dominate which others. The result is a ranking of the bounds in a partial order. It turns out that two of the bounds dominate all others: a bound due to Nelissen and a bound obtained from the linear programming relaxation of a set packing formulation. Experiments show that the latter is almost always optimal and can be computed quickly. Ó 2001 Elsevier Science B.V. All rights reserved.
A Complete Bibliography of Publications in Journal of Computational Chemistry: 1990–1999
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