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15
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.
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... 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. Freem ..."
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Cited by 242 (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
Tight Bounds and 2Approximation Algorithms for Integer Programs with Two Variables per Inequality
 Mathematical Programming
, 1992
"... . The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NPcomplete, even when all variables are binary. This paper deals with integer linear minimization problems in n variables subject to m linear constraints with at most tw ..."
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Cited by 44 (6 self)
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. The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NPcomplete, even when all variables are binary. This paper deals with integer linear minimization problems in n variables subject to m linear constraints with at most two variables per inequality, and with all variables bounded between 0 and U . For such systems, a 2\Gammaapproximation algorithm is presented that runs in time O(mnU 2 log(Un 2 =m)), so it is polynomial in the input size if the upper bound U is polynomially bounded. The algorithm works by finding first a superoptimal feasible solution that consists of integer multiples of 1 2 . That solution gives a tight bound on the value of the minimum. It further more has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem. These properties are a generalization of the properties of the vertex cover problem. The algorithm described is, ...
A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths
, 2000
"... The following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft. Each maintenance period is a job, and the maintenance facilities are machines. In this context, there are very few different types of maintenances performed, so it is natural to consider the ..."
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Cited by 9 (0 self)
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The following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft. Each maintenance period is a job, and the maintenance facilities are machines. In this context, there are very few different types of maintenances performed, so it is natural to consider the problem with only a small, fixed number C of different types of jobs. Each job type has a processing time, and each machine is available for the same length of time. A machine can handle at most one job at a time, all jobs are released at time zero, there are no due dates or precedence constraints, and preemption is not allowed. The question is whether it is possible to finish all jobs. We call this problem the Multiprocessor Scheduling Problem with C job lengths (MSPC). Scheduling problems such as MSPC where we can partition the jobs into a relatively few types such that all jobs of a certain type are identical are often called highmultiplicity problems. Highmultiplicity problems are interesting because their input is very compact: the input to MSPC consists of only 2C + 2 numbers. For the case C = 2 we present a polynomialtime algorithm. We show that this algorithm guarantees a schedule that uses the minimum possible number of different onemachine schedules, namely three. Further, we extend this algorithm to the case of machinedependent deadlines (uniform parallel machines), to a multiparametric case (that contains the case of unrelated parallel machines), and to some related covering problems. Finally, we give some counterexamples showing why our results don’t extend to the case C > 2.
Integer programming
 Handbook of Algorithms and Theory of Computing
, 1999
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On the exact separation of mixed integer knapsack cuts
 Proceedings of the 2007 Integer Programming and Combinatorial Optimization conference
, 2007
"... During the last decades, much research has been conducted deriving classes of valid inequalities for singlerow mixed integer programming polyhedrons. However, no such class has had as much practical success as the MIR inequality when used in cutting plane algorithms for general mixed integer progra ..."
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During the last decades, much research has been conducted deriving classes of valid inequalities for singlerow mixed integer programming polyhedrons. However, no such class has had as much practical success as the MIR inequality when used in cutting plane algorithms for general mixed integer programming problems. In this work we analyze this empirical observation by developing an algorithm which takes as input a point and a mixed integer knapsack polyhedron, and either proves the point is in the convex hull of said polyhedron, or finds a separating hyperplane, or knapsack cut. The main feature of this algorithm is a specialized subroutine for solving the Mixed Integer Knapsack Problem which exploits dominance relationships. To our knowledge, this is the first algorithm proposed for this problem. Exactly separating over the closure of mixed integer knapsack sets allows us to establish natural benchmarks by which to evaluate specific classes of knapsack cuts. Using these benchmarks on Miplib 3.0 and Miplib 2003 instances we analyze the performance of MIR inequalities. Our computations, which are performed in exact arithmetic, are surprising: Averaging over the 78 instances in which knapsack cuts afford bound improvements, MIR cuts alone achieve 95 % of the observed gain. 1
Fast 2Variable Integer Programming
 Integer Programming and Combinatorial Optimization, IPCO 2001, volume 2081 of LNCS
, 2001
"... We show that a 2variable integer program defined by m constraints involving coefficients with at most s bits can be solved with O(m+s log m) arithmetic operations or with O(m+logm log s)M(s) bit operations, where M(s) is the time needed for sbit integer multiplication. ..."
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We show that a 2variable integer program defined by m constraints involving coefficients with at most s bits can be solved with O(m+s log m) arithmetic operations or with O(m+logm log s)M(s) bit operations, where M(s) is the time needed for sbit integer multiplication.
A linear algorithm for Integer programming in the Plane
"... We show that a 2variable integer program, defined by m constraints involving coefficients with at most ϕ bits can be solved with O(m+ϕ) arithmetic operations on rational numbers of size O(ϕ). ..."
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Cited by 4 (0 self)
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We show that a 2variable integer program, defined by m constraints involving coefficients with at most ϕ bits can be solved with O(m+ϕ) arithmetic operations on rational numbers of size O(ϕ).