Greedy Algorithms for Stochastic Scheduling on Unrelated Machines - PowerPoint PPT Presentation

Greedy Algorithms for Stochastic Scheduling on Unrelated Machines Marc Uetz m.uetz@utwente.nl joint work with Varun Gupta Ben Moseley Qiaomin Xie

This Talk Analyses of algorithms stochastic scheduling: 1. either offline 2. or restricted to identical machines 3. or required (sophisticated) LP-relaxations this talk first performance bounds for simple greedy algorithm: - online - unrelated machines - stochastic jobs Marc Uetz - Greedy Alg. for Stochastic Scheduling 2

1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks Marc Uetz - Greedy Alg. for Stochastic Scheduling 3

Single Machine Scheduling Given: n jobs j , weights w j > 0, nonpreemptive processing times p j ∈ Z > 0 Task: sequence jobs on 1 machine; one job at a time; minimize � j w j C j where C j = j ’s completion time; 0 time C red C green C orange C blue Theorem (Smith 1956) Smith’s rule, sequencing jobs in order w j / p j ց is optimal Marc Uetz - Greedy Alg. for Stochastic Scheduling 4

Identical Parallel Machine Scheduling Given: n jobs as above; m identical parallel machines Task: schedule each job on any one machine; minimize � j w j C j 0 time Theorem Problem is strongly NP-hard [Garey & Johnson, Problem SS13] Smith’s rule: tight 1.21-approximation [Kawaguchi & Kyan, 1986] There exists a PTAS [Skutella & Woeginger, 2000] Marc Uetz - Greedy Alg. for Stochastic Scheduling 5

Unrelated Machine Scheduling Given: m machines, machine-dependent processing times p ij Task: schedule each job on one machine; minimize � j w j C j 0 time Theorem Problem is APX-hard [Hoogeveen et al., 2002] Exists ( 3 2 − c ) -approximation [Bansal, Srinivasan, Svensson, 2016] Marc Uetz - Greedy Alg. for Stochastic Scheduling 6

Uncertainty in Scheduling Uncertainty of job sizes p j (or p ij ): non-clairvoyant online models, w. preemption allowed [many, many references] no preemption: non-clairvoyant Ω( n )-competitive Here: no preemption, but with probabilistic info on p j Marc Uetz - Greedy Alg. for Stochastic Scheduling 7

1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks Marc Uetz - Greedy Alg. for Stochastic Scheduling 8

Stochastic Scheduling job size = (independent) random variables P j (or P ij ); all known Pr[ P j ≥ t ] 1 0 time t Solution: Non-anticipatory scheduling policy Π Decisions based on information up to now and a priori knowledge about P j (or P ij ); no further information about the future. 0 time now Marc Uetz - Greedy Alg. for Stochastic Scheduling 9

Optimality Π( I ) := cost of policy Π on instance I , is a random variable Definition (Optimal Policy) Call Π OPT optimal if it achieves inf { E [Π( I )] | Π non-anticipatory policy } Marc Uetz - Greedy Alg. for Stochastic Scheduling 10

Example n = 4 jobs, weights w j = 1 time 0 1 10 blue jobs: P j = 1 � 0 probability 4/5 green jobs: P j = probability 1/5 (note E [ P j ] = 2) 10 Schedule on m = 2 identical machines. Marc Uetz - Greedy Alg. for Stochastic Scheduling 11

Stochastic World Unique optimal policy: Start green + blue � green → blue if first green job short Then continue: blue → green if first green job long [with Π( I ) = E [ � j C j ] = 6 . 76]. 0 1 2 10 11 12 Complicated tradeoff between large E [ P j ] or large Pr( P j = “ ∞ ”) (i.e., heavy tail) Even deliberate idleness may be necessary [U. 2003] Marc Uetz - Greedy Alg. for Stochastic Scheduling 12

Approximation Algorithms Optimal policies intuitively complex, exponential size decision tree; definitely NP(APX)-hard, . . . only computing E [Π( I )] can be #P-hard [Hagstrom, 1988] Definition (Approximation) Policy Π has performance guarantee α ≥ 1, if for all instances I E [Π( I )] ≤ α E [Π OPT ( I )] Adversary is non-anticipatory, too! Marc Uetz - Greedy Alg. for Stochastic Scheduling 13

Approximation Algorithms M¨ ohring, Schulz & U. [JACM, 1999] First LP-based approximation algorithms e.g.: Smith’s rule has performance guarantee ( 3+∆ 2 ). Skutella & U. [SICOMP, 2005] , Megow, U. & Vredeveld [Math. OR, 2006] Chou et al. [OR 2006], Schulz [2008] Problems w. precedence constraints, jobs that arrive online. Moseley, Im, Pruhs [STACS, 2015] O( log 2 n + m log n )-approximation All results ↑ for identical machines Skutella, Sviridenko, U. [Math. OR, 2016] Guarantee ( 3+∆ 2 ) for unrelated machines, time-indexed LP Marc Uetz - Greedy Alg. for Stochastic Scheduling 14

1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks Marc Uetz - Greedy Alg. for Stochastic Scheduling 15

Main Result Model 1: Stochastic jobs appear one after another ( t = 0), must be assigned to machine upon arrival Theorem Greedy online algorithm has a performance guarantee (8 + 4∆) . ∆ = upper bound on the (squared) coeff. of variation CV [ P ij ] := V ar [ P ij ] / E 2 [ P ij ] ≤ ∆ for all P ij Model 2: stochastic jobs arrive over time (at release times r j ), performance guarantee is (144 + 72∆) Marc Uetz - Greedy Alg. for Stochastic Scheduling 16

Algorithm Algorithm Greedy per machine, jobs sequenced as in Smith’s rule ( w j / E [ P ij ]) when job j arrives (order 1 , 2 , . . . ), compute for each machine i ∈ M expected instantaneous increase in objective, i.e., � � w j E [ P ij ] + w j E [ P ik ] + E [ p ij ] w k . k < j , k → i , k ∈ H ( j , i ) k < j , k → i , k ∈ L ( j , i ) assign job j to any machine minimizing this quantity j i H ( j , i ) L ( j , i ) 0 Marc Uetz - Greedy Alg. for Stochastic Scheduling 17

Rough Sketch Analysis 1. set up LP Relaxation (for stochastic problem) 2. simplify LP relaxation – losing O( ∆ ) 3. analysis of Greedy using dual solution – losing O( 1 ) called “dual fitting” by [Anand et al., SODA 2012] Marc Uetz - Greedy Alg. for Stochastic Scheduling 18

1 - Time-Indexed LP Relaxation Instance I and non-anticipatory policy Π, y ijt := Pr[Π has job j in process on machine i at [ t , t + 1)] 0 1 2 10 11 12 second, blue job, j = 4, has y 1 , 4 , 0 = 16/25 y 2 , 4 , 1 = 9/25 Marc Uetz - Greedy Alg. for Stochastic Scheduling 19

1 - Time-Indexed LP Relaxation Instance I and non-anticipatory policy Π, y ijt := Pr[Π has job j in process on machine i at [ t , t + 1)] Properties of y ijt : y ijt � � E [ P ij ] = 1 for all j ∈ J [Π non-anticipatory!] i ∈ M t ≥ 0 � j ∈ J y ijt ≤ 1 for all i ∈ M , t ≥ 0 with some calculus, one shows that � y ijt + 1 − CV [ P ij ] 2 � t + 1 � � E [ C j ] = � � y ijt t ≥ 0 i ∈ M E [ P ij ] 2 2 for all j ∈ J E [ C j ] ≥ � � t ≥ 0 y ijt [for analysis] i ∈ M Marc Uetz - Greedy Alg. for Stochastic Scheduling 20

1 - LP Relaxation (for stochastic problem) z S := min � w j C S j j ∈ J � y ijt + 1 − CV [ P ij ] 2 1 − CV [ P ij ] 2 � � � C S � t + 1 � s.t. j = y ijt 2 E [ P ij ] 2 t ≥ 0 i ∈ M y ijt � � E [ P ij ] = 1 ∀ jobs j , i ∈ M t ≥ 0 � y ijt ≤ 1 ∀ machines i , times t , j ∈ J � � y ijt ≤ C j ∀ jobs j , t ≥ 0 i ∈ M y ijt ≥ 0 ∀ jobs j , machines i , times t . Would like to work with (LP) dual , but. . . Marc Uetz - Greedy Alg. for Stochastic Scheduling 21

2 - Simplified LP Relaxation z P := min � w j C P j j ∈ J � y ijt � + 1 � � C P � t + 1 � s.t. j = 2 y ijt 2 E [ P ij ] t ≥ 0 i ∈ M y ijt � � E [ P ij ] = 1 jobs j , i ∈ M t ≥ 0 � y ijt ≤ 1 machines i , times t , j ∈ J y ijt ≥ 0 jobs j , machines i , times t . Lemma 1 + ∆ z P ≤ z S � � 2 Marc Uetz - Greedy Alg. for Stochastic Scheduling 22

2 - LP Dual the dual has variables ( α, β ); z D = � � � max α j − β is t ≥ 0 j ∈ J i ∈ M � t + 1 2 + E [ P ij ] � s.t. α j ≤ E [ P ij ] β i , t + w j for all i , j , t 2 β it ≥ 0 for all i , t Goal: dual solution that relates to Greedy Marc Uetz - Greedy Alg. for Stochastic Scheduling 23

3 - Greedy & Dual LP Interpreting the Dual α j := expected increase (of Greedy ) of objective upon arrival of job j β i , t := total expected weight of jobs assigned to machine i , and still unfinished at time t (by Greedy ) Can show Lemma Solution ( α/ 2 , β/ 2) is dual feasible. Yet of little help, as z D = � j ∈ J α j − � � t ≥ 0 β i , t , and i ∈ M � j ∈ J α j = objective value of Greedy � � t ≥ 0 β it = objective value of Greedy , too i ∈ M Marc Uetz - Greedy Alg. for Stochastic Scheduling 24

3 - Speed Augmentation Alternative dual LP solution α j ˜ α j := expected increase (of Greedy ) of objective upon arrival of job j β i , t ˜ β i , t := total expected weight of jobs assigned to machine i , and still unfinished at time t (by Greedy ) For the same instance, but assuming all machines run at speed 2. Now can show Lemma α/ 2 , ˜ Solution (˜ β/ 4) is feasible (for original LP) 2 α and ˜ α = 1 Proof: as ˜ β i , t = β i , 2 t . . . by dual constraint. . . Marc Uetz - Greedy Alg. for Stochastic Scheduling 25

3 - Putting Things Together Now we have β/ 4) = 1 α j − 1 z D (˜ α/ 2 , ˜ � � � ˜ ˜ β i , t 2 4 j ∈ J i ∈ M t ≥ 0 = 1 4 Greedy − 1 8 Greedy So β/ 4) ≤ 8 z D = 8 z P α/ 2 , ˜ Greedy = 8 z D (˜ ≤ 8(1 + ∆ 2 ) z S ≤ (8 + 4∆) OPT Marc Uetz - Greedy Alg. for Stochastic Scheduling 26