Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz - PowerPoint PPT Presentation

Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz m.uetz@utwente.nl (updated) IPCO 2017 paper, with: Varun Gupta, Ben Moseley, Qiaomin Xie

1-Slide Overview Talk is about (basic, notorious) stochastic scheduling problem [defined later] Almost all previous results (scheduling in general): good algorithms based on solving (complicated LP) relaxations Alternative title: To hell with (LP-)relaxations! if greedy (online) algorithms are good enough first results (for this model) where jobs appear online Greedy algorithm has performance guarantee (144 + 72∆)(72 + 36∆)(12 + 6∆)(6 + 3∆) h (∆) [∆ = (squared) coeff. variation, 1 ≤ h ( · ) ≤ 2 with h (0) = 1 ] Marc Uetz - Greed. . . is good 2

“Greed, . . . , greed is good!” Marc Uetz - Greed. . . is good 3

Unrelated Machine Scheduling – R | | � w j C j Given: m machines, n non-preemptive jobs with weights w j and machine-dependent processing times p ij : C j := completion time job j in schedule; minimize � j w j C j 0 C blue time Theorem [Hoogeveen et al., 2002] Offline: problem is APX-hard Offline: LP-based 1 . 4994 -approximation [Li, 2018] Online: lower bound 1 . 309 [Vestjens 1997] Marc Uetz - Greed. . . is good 4

Uncertainty in Scheduling Job j appears at release time r j Non-clairvoyant online models (job lengths unknown till C j ) - hopeless, since jobs cannot be preempted [Ω( n ) even in simplest settings] Clairvoyant online models (job lengths known upon arrival r j ) - deterministic algo.: 8-competitive [Hall et al. MOR 1997] - randomized algo.: 5.78-competitive [Chakrabarti et al. ICALP 1996] Stochastic Online Model (= Data Science) Non-clairvoyant, but probabilistic info on p ij [our result ⇒ Greedy is deterministic 6-competitive algo.] Marc Uetz - Greed. . . is good 5

1 Stochastic Scheduling 2 Greedy Algorithm for Unrelated Machines 3 Final Remarks Marc Uetz - Greed. . . is good 6

Stochastic Processing Times job j appears: size = (independent) random variables P ij ; known Pr[ P ij ≥ t ] 1 0 time t Solution: Non-anticipatory scheduling policy Π Decisions may only use information up to now. 0 time now Objective: Min. expected performance E [ � w j C j ] = � w j E [ C j ] Marc Uetz - Greed. . . is good 7

Example with Four Jobs n = 4 jobs, all weights w j = 1 time 0 1 10 blue jobs: P j = 1 � 0 probability 4/5 green jobs: P j = probability 1/5 ( E [ P j ] = 2) 10 Optimal policy for m = 2 identical machines? Marc Uetz - Greed. . . is good 8

Policies are Complex (and Dynamic) Unique optimal policy: Start green + blue � first green then blue if first green job = 0 Then continue: first blue then green if first green job = 10 [with E [ � j w j C j ] = 6 . 76] 0 1 2 10 11 12 Complicated tradeoff between small E [ P j ] or large Pr( P j = 0) (but, heavy tail) Marc Uetz - Greed. . . is good 9

Approximation for Stochastic Scheduling Optimal policies hopeless, even offline, . . . Definition (Approximation) Policy Π has performance guarantee α ≥ 1, if for all instances P � � w j C Π w j C OPT E [ j ] ≤ α E [ ] j Adversary OPT knows set of jobs, but subject to uncertain processing times P ij , too Classical competitive analysis is special case Marc Uetz - Greed. . . is good 10

Coefficient of Variation Performance guarantees depend on “variability” of P ij Define ∆ := max i , j CV [ P ij ] 2 = V ar [ P ij ] / E 2 [ P ij ] Marc Uetz - Greed. . . is good 11

Approximation Algorithms Stochastic Scheduling For identical machines: M¨ ohring, Schulz & U. [J.ACM, 1999] first LP-based approximation algorithm e.g.: Smith’s rule ( w j / E [ P j ] ց ) has guarantee ( 3+∆ 2 ) √ improved to 1 + 1 2 ( 2 − 1)(1 + ∆) [J¨ ager & Skutella 2018] Skutella & U. [SICOMP, 2005] , Megow, U. & Vredeveld [Math. OR, 2006] Chou et al. [OR 2006], Schulz [COCOA 2008] Problems w. precedence constraints, release times, or online Im, Moseley, Pruhs [STACS, 2015] remarkable O( log 2 n + m log n )-approximation For unrelated machines: Skutella, Sviridenko, U. [Math. OR, 2016] ( 3+∆ 2 ) for offline , using time-indexed LP relaxation Marc Uetz - Greed. . . is good 12

Greedy for Unrelated Machines, Stochastic Jobs Theorem If all release times r j = 0 (“online-list”), Greedy has performance guarantee 4 + 2∆ , analysis tight for ∆ = 0 [Correa & Queyranne, 2012] . Theorem In general (release times r j > 0, “online-time”), Greedy has performance guarantee (6 + 3∆) h (∆) . 1 ≤ h ( · ) ≤ 2 with h (0) = 1 , h (1) = 3 / 2 Marc Uetz - Greed. . . is good 13

Greedy Algorithm I sequencing: available jobs w. max. w j / E [ P ij ] go first assignment: use “proxy” for E [increase] of objective: job j appears ar r j , consider all jobs that arrived earlier, their machine assignments fixed, and assuming p ij = E [ P ij ] and r k = 0 for all jobs, compute expected increase of � j w j E [ C j ] if j was inserted into sequence in order of w j / E [ P ij ] → assign job j to any machine minimizing this increase j i 0 Marc Uetz - Greed. . . is good 14

Greedy Algorithm II sequencing: available jobs w. max. w j / E [ P ij ] go first availability: job j declared available on machine i at slightly inflated release time r ij ≥ r j r ij = max { E [ P ij ] , s j } where s j = start time of job j in nominal Greedy schedule where we assume p ik = E [ P ik ] for all jobs on i [note: this can be computed online] j i r j r ij 0 Marc Uetz - Greed. . . is good 15

Sketch Analysis 1. time-indexed LP Relaxation (for stochastic problem) - sorry 2. “simplify” that LP relaxation – losing O( ∆ ) 3. analysis of Greedy using dual LP solution – losing O( 1 ) “dual fitting” [Anand et al., SODA 2012] Marc Uetz - Greed. . . is good 16

1 - Time-Indexed LP y ijt := Pr[machine i has job j in process at [ t , t + 1)] 0 1 2 10 11 12 second, green job (say j = 3), has at time t = 2 y 1 , 3 , 2 = 4/25 = 1/25 y 2 , 3 , 2 Marc Uetz - Greed. . . is good 17

1 - LP Relaxation (Not a Formulation!) � 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 i ∈ M t ≥ 0 y ijt � � E [ P ij ] = 1 ∀ jobs j , i ∈ M t ≥ r j � y ijt ≤ 1 ∀ machines i , times t , j ∈ J � � y ijt ≤ C S ∀ jobs j , j i ∈ M t ≥ r j y ijt ≥ 0 ∀ jobs j , machines i , times t . Would like to work with (LP) dual , but. . . Marc Uetz - Greed. . . is good 18

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 ] i ∈ M t ≥ 0 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 - Greed. . . is good 19

2 - LP Dual the dual has variables ( α, β ); � � � z D = max α j − β it j ∈ J i ∈ M t ≥ 0 � � t + E [ P ij ] + 1 s.t. α j ≤ E [ P ij ] β it + w j for all i , j , t 2 2 β it ≥ 0 for all i , t Lemma Considering Greedy det (for p ij = E [ p ij ]); can construct feasible α, ˜ dual solution (˜ β ) with β ) = 1 z D (˜ α, ˜ 6 Greedy det [idea: 3 x speed augmentation] Marc Uetz - Greed. . . is good 20

3 - Final Steps By duality Greedy det = 6 z D (˜ α, ˜ β ) ≤ 6 z D =6 z P ≤ 6(1 + ∆ 2 ) z S ≤ (6 + 3∆) OPT Finally, can show for (true) expected starting time job j Lemma s det E [ S j ] ≤ h (∆) j � �� � ≤ 2 s det = starting time job j in Greedy det j Proof: � ( P k − E [ P k ]) + . . . S j ≤ s j + predecessors k Marc Uetz - Greed. . . is good 21

Final Remarks O( ∆ ) is tight (for greedy), there is a ∆ / 2 lower bound open problems 1. is const. approximation (indep. of ∆) possible? 2. is stochastic problem harder to approximate ? thanks for your attention! Marc Uetz - Greed. . . is good 22