Stochastic Load Balancing on Unrelated Machines Viswanath Nagarajan Industrial & Operations Engineering Department University of Michigan Joint work with Anupam Gupta (CMU), Amit Kumar (IIT Delhi), Xiangkun Shen (UM) V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 1 / 25
Outline Introduction 1 Motivation Related Work Results Techniques 2 Effective Size Algorithm Conclusion 3 V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 2 / 25
Load Balancing Problem Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66] . AB ACD D BC ABCD D AC A B C D V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 3 / 25
Load Balancing Problem Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66] . AB ACD D BC ABCD D AC A B C D V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 3 / 25
Load Balancing Problem Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66] . AB ACD D BC ABCD D AC A B C D V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 3 / 25
Load Balancing Problem Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66] . AB ACD D BC ABCD D AC A B C D V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 3 / 25
Load Balancing Problem Schedule jobs on machines to minimize the maximum load. One of the first approximation algorithms devised [Graham 66] . AB ACD D BC ABCD D AC A makespan B C D V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 3 / 25
Load Balancing (Formally) n jobs and m machines. p ij = size of job j on machine i . Identical machines: p ij = p j . Related machines: ( p ij ) matrix rank 1. Unrelated machines: general case. Find an assignment to minimize makespan: m � min max p ij . J 1 , ··· J m i =1 j ∈ J i J i = set of jobs assigned to machine i . V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 4 / 25
Optimization under Uncertainty In many situations precise input data unknown. Various models to deal with uncertainty. Stochastic : input drawn from some distribution. Robust : input drawn from some uncertainty set. Online : no prior information about input. V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 5 / 25
Optimization under Uncertainty In many situations precise input data unknown. Various models to deal with uncertainty. Stochastic : input drawn from some distribution. Robust : input drawn from some uncertainty set. Online : no prior information about input. Here: stochastic setting. V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 5 / 25
Stochastic Load Balancing n jobs and m machines. Random variable X ij is size of job j on machine i . Arbitrary distributions. Independent and known upfront. Find an assignment { J i } m i =1 to minimize expected makespan: m � . max E X ij i =1 j ∈ J i ? ? ? ? ? V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 6 / 25
Stochastic Load Balancing n jobs and m machines. Random variable X ij is size of job j on machine i . Arbitrary distributions. Independent and known upfront. Find an assignment { J i } m i =1 to minimize expected makespan: m � . max E X ij i =1 j ∈ J i ? ? ? ? ? V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 6 / 25
Stochastic Load Balancing n jobs and m machines. Random variable X ij is size of job j on machine i . Arbitrary distributions. Independent and known upfront. Find an assignment { J i } m i =1 to minimize expected makespan: m � . max E X ij i =1 j ∈ J i ? ? ? ? ? ? ? ? ? ? V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 6 / 25
Stochastic Load Balancing n jobs and m machines. Random variable X ij is size of job j on machine i . Arbitrary distributions. Independent and known upfront. Find an assignment { J i } m i =1 to minimize expected makespan: m � . max E X ij i =1 j ∈ J i ? ? ? ? ? V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 6 / 25
Stochastic Load Balancing n jobs and m machines. Random variable X ij is size of job j on machine i . Arbitrary distributions. Independent and known upfront. Find an assignment { J i } m i =1 to minimize expected makespan: m � . max E X ij i =1 j ∈ J i ? ? ? ? ? minimize expected makespan V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 6 / 25
Natural Approach for Stochastic Optimization 1 Replace each random variable X in the stochastic problem by deterministic surrogate d ( X ). 2 Solve the resulting deterministic problem. 3 Return the same solution for the stochastic problem. V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 7 / 25
Deterministic Surrogate for Load Balancing? Expected size? V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 8 / 25
Deterministic Surrogate for Load Balancing? Expected size? Type 1: size 1 (deterministic). 1 Type 2: size ∼ (0 , 1) Bernoulli r.v. with p = √ m . m − √ m machines m − √ m machines log m E [ mkspan ] = m − √ m jobs: expectation 1 √ m log log m √ m machines √ m 1 m jobs: expectation √ m V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 8 / 25
Deterministic Surrogate for Load Balancing? Expected size? Type 1: size 1 (deterministic). 1 Type 2: size ∼ (0 , 1) Bernoulli r.v. with p = √ m . m − √ m machines m − √ m machines log m E [ mkspan ] = m − √ m jobs: expectation 1 √ m log log m √ m machines √ m 1 m jobs: expectation √ m V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 8 / 25
Deterministic Surrogate for Load Balancing? Expected size? Type 1: size 1 (deterministic). 1 Type 2: size ∼ (0 , 1) Bernoulli r.v. with p = √ m . m − √ m machines m − √ m machines m − √ m machines m − √ m machines log m E [ mkspan ] ≤ 2 E [ mkspan ] = √ m log log m √ m machines √ m machines √ m 1 m jobs: expectation √ m V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 8 / 25
Deterministic Surrogate for Load Balancing? Expected size? Type 1: size 1 (deterministic). 1 Type 2: size ∼ (0 , 1) Bernoulli r.v. with p = √ m . m − √ m machines m − √ m machines m − √ m machines m − √ m machines log m E [ mkspan ] ≤ 2 E [ mkspan ] = √ m log log m √ m machines √ m machines √ m 1 m jobs: expectation √ m V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 8 / 25
Deterministic Surrogate for Load Balancing? Expected size? Type 1: size 1 (deterministic). 1 Type 2: size ∼ (0 , 1) Bernoulli r.v. with p = √ m . m − √ m machines m − √ m machines m − √ m machines m − √ m machines log m E [ mkspan ] ≤ 2 E [ mkspan ] = √ m log log m √ m machines √ m machines √ m 1 m jobs: expectation √ m Effective size [Hui 88] [Elwalid, Mitra 93] [Kelly 96] [Kleinberg, Rabani, Tardos 00] . V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 8 / 25
Outline Introduction 1 Motivation Related Work Results Techniques 2 Effective Size Algorithm Conclusion 3 V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 9 / 25
Related Work: Deterministic Job Sizes Identical machines: ◮ List Scheduling: 2-approximation [Graham 66] . ◮ Sorted List Scheduling: 4 3 -approximation [Graham 69] . ◮ PTAS [Hochbaum, Shmoys 87] . Related machines: ◮ 2-approximation [Morrison 88] [Gonzalez, Ibarra, Sahni 77] . ◮ PTAS [Hochbaum, Shmoys 88] . Unrelated machines: ◮ 2-approximation [Lenstra, Shmoys, Tardos 90] . ◮ APX-hardness [Shmoys, Tardos 93] . ◮ 2 − 1 6 + ǫ -approximation for restricted assignment [Jansen, Rohwedder 17] . V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 10 / 25
Related Work: Stochastic Job Sizes O (1)-approximation for identical machines [Kleinberg, Rabani, Tardos 00] . #P-hard to evaluate objective Better results for special classes of job size distributions [Goel, Indyk 99] . ◮ Poisson distributed job sizes: 2-approximation. ◮ Exponential distributed job sizes: PTAS. V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 11 / 25
Related Work: Stochastic Job Sizes O (1)-approximation for identical machines [Kleinberg, Rabani, Tardos 00] . #P-hard to evaluate objective Better results for special classes of job size distributions [Goel, Indyk 99] . ◮ Poisson distributed job sizes: 2-approximation. ◮ Exponential distributed job sizes: PTAS. Related knapsack and bin-packing problems [Deshpande, Li 11] . Other models for stochastic combinatorial optimization: Two-stage [Shmoys, Swamy 04] [Gupta, Pal, Ravi, Sinha 04] .. Adaptive [Dean Goemans Vondrak 08] [Guha Munagala 07] [Bhalgat Goel Khanna 11] .. V. Nagarajan (UM) Stochastic Load Balancing Aussois, 2019 11 / 25
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