Fair Scheduling via Iterative Quasi-Uniform Sampling

Im, Sungjin; Moseley, Benjamin

doi:10.1137/1.9781611974782.171

Cited by 4 publications

(5 citation statements)

References 24 publications

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“…In the special case of stretch metric, where w j = 1/p j , PTAS is known [6,9]. The problem of minimizing (unweighted) ℓ p norm of flow-times was studied by Im and Moseley [12] who gave a constant factor approximation in polynomial time.…”

Section: Related Workmentioning

confidence: 99%

Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-Polynomial Time

Batra

Garg

Kumar

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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In the weighted flow-time problem on a single machine, we are given a set of n jobs, where each job has a processing requirement p j , release date r j and weight w j . The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudo-polynomial time constant approximation algorithm for this problem. The algorithm also extends directly to the problem of minimizing the ℓ p norm of weighted flow-times. The running time of our algorithm is polynomial in n, the number of jobs, and P , which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multi-cut problem on trees, which we call Demand MultiCut problem. Even though we do not give a constant factor approximation algorithm for the Demand MultiCut problem on trees, we show that the specific instances of Demand MultiCut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of DP table.

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Section: Related Workmentioning

confidence: 99%

Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-Polynomial Time

Batra

Garg

Kumar

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…The cost is as if only one iteration of uniform sampling occurred. This is similar to the analysis approach used in quasi-uniform sampling [23] and rounding [13].…”

Section: The Rounding Algorithmmentioning

confidence: 88%

“…The modification ensures that (1) in expectation no variable, over all iterations, is sampled by more than a O(log log nP ) factor than how much it is selected by the optimal LP solution and (2) the relaxation remains feasible. The scheme builds on techniques of quasi-uniform sampling [4,23] and quasi-uniform iterative rounding [13].…”

Section: Overview Of Technical Contributionsmentioning

confidence: 99%

Scheduling to Approximate Minimization Objectives on Identical Machines

Moseley¹

2019

Preprint

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This paper considers scheduling on identical machines. The scheduling objective considered in this paper generalizes most scheduling minimization problems. In the problem, there are n jobs and each job j is associated with a monotonically increasing function g j . The goal is to design a schedule that minimizes j∈[n] g j (C j ) where C j is the completion time of job j in the schedule. An O(1)-approximation is known for the single machine case. On multiple machines, this paper shows that if the scheduler is required to be either non-migratory or non-preemptive then any algorithm has an unbounded approximation ratio. Using preemption and migration, this paper gives a O(log log nP )-approximation on multiple machines, the first result on multiple machines. These results imply the first non-trivial positive results for several special cases of the problem considered, such as throughput minimization and tardiness.Natural linear programs known for the problem have a poor integrality gap. The results are obtained by strengthening a natural linear program for the problem with a set of covering inequalities we call job cover inequalities. This linear program is rounded to an integral solution by building on quasi-uniform sampling and rounding techniques.

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“…For identical release times, [10] gave an improved 4+ε polynomial time approximation, and [1] gave a quasi-polynomial time approximation scheme. For general release times, better O(1) approximation guarantees were obtained for various important objective functions such as k norm of flow times [13] and weighted flow times [5,11]. The general scheduling problem has also been considered in the online setting [14].…”

Section: Introductionmentioning

confidence: 99%

“…He considers a time-indexed LP formulation strengthened by certain job-cover and knapsack-cover inequalities. Using various structural properties of the LP, he applies the quasi-uniform sampling technique [16,9,13] in a clever way to round this LP and obtain an O(log log nP ) approximation.…”

Section: Introductionmentioning

confidence: 99%

Non-uniform Geometric Set Cover and Scheduling on Multiple Machines

Bansal¹,

Batra²

2019

Preprint

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We consider the following general scheduling problem: there are m identical machines and n jobs all released at time 0. Each job j has a processing time p j , and an arbitrary non-decreasing function f j that specifies the cost incurred for j, for each possible completion time. The goal is to find a preemptive migratory schedule of minimum cost. This models several natural objectives such as weighted norm of completion time, weighted tardiness and much more.We give the first O(1) approximation algorithm for this problem, improving upon the O(log log nP ) bound due to Moseley (2019). To do this, we first view the job-cover inequalities of Moseley geometrically, to reduce the problem to that of covering demands on a line by rectangular and triangular capacity profiles. Due to the non-uniform capacities of triangles, directly using quasi-uniform sampling loses a O(log log P ) factor, so a second idea is to adapt it to our setting to only lose an O(1) factor. Our ideas for covering points with non-uniform capacity profiles (which have not been studied before) may be of independent interest.

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Fair Scheduling via Iterative Quasi-Uniform Sampling

Cited by 4 publications

References 24 publications

Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-Polynomial Time

Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-Polynomial Time

Scheduling to Approximate Minimization Objectives on Identical Machines

Non-uniform Geometric Set Cover and Scheduling on Multiple Machines

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