# On the Value of Job Migration in Online Makespan Minimization

On the Value of Job Migration in Online Makespan Minimization Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of n jobs has to be scheduled non-preemptively on m machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range [1.88, 1.9201]. Currently no randomized online algorithm with a smaller competitiveness is known, for general m. In this paper we explore the power of job migration, i.e. an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is $$\alpha _m$$ α m -competitive, for any $$m\ge 2$$ m ≥ 2 , where $$\alpha _m$$ α m is the solution of a certain equation. For $$m=2$$ m = 2 , $$\alpha _2 = 4/3$$ α 2 = 4 / 3 and $$\lim _{m\rightarrow \infty } \alpha _m = W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2)) \approx 1.4659$$ lim m → ∞ α m = W - 1 ( - 1 / e 2 ) / ( 1 + W - 1 ( - 1 / e 2 ) ) ≈ 1.4659 . Here $$W_{-1}$$ W - 1 is the lower branch of the Lambert W function. For $$m\ge 11$$ m ≥ 11 , the algorithm uses at most 7m migration operations. For smaller m, 8m to 10m operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses o(n) job migrations can achieve a competitive ratio smaller than $$\alpha _m$$ α m . We finally trade performance for migrations. We give a family of algorithms that is c-competitive, for any $$5/3\le c \le 2$$ 5 / 3 ≤ c ≤ 2 . For $$c= 5/3$$ c = 5 / 3 , the strategy uses at most 4m job migrations. For $$c=1.75$$ c = 1.75 , at most 2.5m migrations are used. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algorithmica Springer Journals

# On the Value of Job Migration in Online Makespan Minimization

, Volume 79 (2) – Sep 8, 2016
26 pages

/lp/springer_journal/on-the-value-of-job-migration-in-online-makespan-minimization-0HmZgLxy0q
Publisher
Springer US
Subject
Computer Science; Algorithm Analysis and Problem Complexity; Theory of Computation; Mathematics of Computing; Algorithms; Computer Systems Organization and Communication Networks; Data Structures, Cryptology and Information Theory
ISSN
0178-4617
eISSN
1432-0541
D.O.I.
10.1007/s00453-016-0209-9
Publisher site
See Article on Publisher Site

### Abstract

Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of n jobs has to be scheduled non-preemptively on m machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range [1.88, 1.9201]. Currently no randomized online algorithm with a smaller competitiveness is known, for general m. In this paper we explore the power of job migration, i.e. an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is $$\alpha _m$$ α m -competitive, for any $$m\ge 2$$ m ≥ 2 , where $$\alpha _m$$ α m is the solution of a certain equation. For $$m=2$$ m = 2 , $$\alpha _2 = 4/3$$ α 2 = 4 / 3 and $$\lim _{m\rightarrow \infty } \alpha _m = W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2)) \approx 1.4659$$ lim m → ∞ α m = W - 1 ( - 1 / e 2 ) / ( 1 + W - 1 ( - 1 / e 2 ) ) ≈ 1.4659 . Here $$W_{-1}$$ W - 1 is the lower branch of the Lambert W function. For $$m\ge 11$$ m ≥ 11 , the algorithm uses at most 7m migration operations. For smaller m, 8m to 10m operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses o(n) job migrations can achieve a competitive ratio smaller than $$\alpha _m$$ α m . We finally trade performance for migrations. We give a family of algorithms that is c-competitive, for any $$5/3\le c \le 2$$ 5 / 3 ≤ c ≤ 2 . For $$c= 5/3$$ c = 5 / 3 , the strategy uses at most 4m job migrations. For $$c=1.75$$ c = 1.75 , at most 2.5m migrations are used.

### Journal

AlgorithmicaSpringer Journals

Published: Sep 8, 2016

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