A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach
01 Jun 1988-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 17, Iss: 3, pp 539-551
TL;DR: A family of polynomial-time algorithms are given such that the last job to finish is completed as quickly as possible and the algorithm delivers a solution that is within a relative error of the optimum.
Abstract: In this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors. More specifically, the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible. We give a family of polynomial-time algorithms {A∈} such that A∈ delivers a solution that is within a relative error of e of the optimum. The technique employed is the dual approximation approach, where infeasible but superoptimal solutions for a related (dual) problem are converted to the desired feasible but possibly suboptimal solution.
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TL;DR: A taxonomy that classifies 27 scheduling algorithms and their functionalities into different categories is proposed, with each algorithm explained through an easy-to-understand description followed by an illustrative example to demonstrate its operation.
Abstract: Static scheduling of a program represented by a directed task graph on a multiprocessor system to minimize the program completion time is a well-known problem in parallel processing. Since finding an optimal schedule is an NP-complete problem in general, researchers have resorted to devising efficient heuristics. A plethora of heuristics have been proposed based on a wide spectrum of techniques, including branch-and-bound, integer-programming, searching, graph-theory, randomization, genetic algorithms, and evolutionary methods. The objective of this survey is to describe various scheduling algorithms and their functionalities in a contrasting fashion as well as examine their relative merits in terms of performance and time-complexity. Since these algorithms are based on diverse assumptions, they differ in their functionalities, and hence are difficult to describe in a unified context. We propose a taxonomy that classifies these algorithms into different categories. We consider 27 scheduling algorithms, with each algorithm explained through an easy-to-understand description followed by an illustrative example to demonstrate its operation. We also outline some of the novel and promising optimization approaches and current research trends in the area. Finally, we give an overview of the software tools that provide scheduling/mapping functionalities.
1,373 citations
TL;DR: This paper considers the question of determining whether a function f has property P or is ε-far from any function with property P, and devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a clique of density p-Clique with respect to the vertex set.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.
1,027 citations
Cites background from "A polynomial approximation scheme f..."
..., [HOCHBAUM AND SHMOYS 1987; HOCHBAUM AND SHMOYS 1988])....
[...]
...RELATION TO DUAL APPROXIMATION (CF., [HOCHBAUM AND SHMOYS 1987; HOCHBAUM AND SHMOYS 1988])....
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TL;DR: It is proved that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP, and a complexity classification for all special cases with a fixed number of processing times is obtained.
Abstract: We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep
ij
. The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.
953 citations
Cites background from "A polynomial approximation scheme f..."
...For any fixed number of machines, Horowitz and Sahni (1976) presented a fully polynomial approximation scheme, whereas Hochbaum and Shmoys (1988) gave a polynomial approximation scheme for the case that the number of machines is a part of the problem instance....
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Journal Article•
TL;DR: In this paper, the authors consider the question of determining whether a function f has property P or is e-far from any function with property P. In some cases, it is also allowed to query f on instances of its choice.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.
870 citations
TL;DR: This note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint and one of the algorithms presented here is stronglyPolynomial.
Abstract: This note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint. The main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value, whose ratio is polynomially bounded, are known. In spite of this difficulty, one of the algorithms presented here is strongly polynomial. Applications to other problems are also discussed.
590 citations
References
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01 Jan 1979
42,654 citations
01 Jan 1979
12,336 citations
TL;DR: In this paper, precise bounds are derived for several anomalies of this type in a multiprocessing system composed of many identical processing units operating in parallel, and they show that an increase in the number of processing units can cause an increased total length of time needed to process a fixed set of tasks.
Abstract: It is known that in multiprocessing systems composed of many identical processing units operating in parallel, certain timing anomalies may occur; e.g., an increase in the number of processing units can cause an increase in the total length of time needed to process a fixed set of tasks. In this paper, precise bounds are derived for several anomalies of this type.
1,449 citations
TL;DR: Exact and approximate algorithms are presented for scheduling independent tasks in a multiprocessor environment in which the processors have different speeds and are guaranteed to obtain solutions that are close to the optimal.
Abstract: Exact and approximate algorithms are presented for scheduling independent tasks in a multiprocessor environment in which the processors have different speeds. Dynamic programming type algorithms are presented which minimize finish time and weighted mean flow time on two processors. The generalization to m processors is direct. These algorithms have a worst-case complexity which is exponential in the number of tasks. Therefore approximation algorithms of low polynomial complexity are also obtained for the above problems. These algorithms are guaranteed to obtain solutions that are close to the optimal. For the case of minimizing mean flow time on m-processors an algorithm is given whose complexity is O(n log mn).
452 citations
TL;DR: This work studies the performance of LPT (largest processing time) schedules with respect to optimal schedules in a nonpreemptive multiprocessor environment to find out if the tasks being scheduled are independent.
Abstract: We study the performance of LPT (largest processing time) schedules with respect to optimal schedules in a nonpreemptive multiprocessor environment. The processors are assumed to have different speeds and the tasks being scheduled are independent.
292 citations