Approximation algorithms for stochastic sequencing and scheduling problems

1. Approximation algorithms for stochastic sequencing and scheduling problems David B. Shmoys Cornell University

2. Worst-Case vs Stochastic Analysis • Yin and Yang of algorithmic analysis • Early results always focus on simple problems for which simple algorithms work • Examples Smith Ratio Rule for 1|| wj Cj Rothkopf WSEPT rule for 1||E[  wjCj ] • Stochastic literature focus on classes of distributions where nice characterizations could be proved e.g. Weiss & Pinedo (exponential distributions play a central role in this line of research)

3. Advent of Competitive Analysis • Sleator & Tarjan for paging problems • Redefined the meaning of on-line more generally to be computing in the absence of full information • On-line scheduling used to mean just jobs arrived over time and existence of job is known only at its release date • Now, many other on-line scheduling models exist • But are these models too pessimistic?

4. Approximation for Stochastic Models • Trend to make models less distribution specific • But now search for settings in which constant performance guarantees can be found • Several examples from a range of types of scheduling and sequencing problems • Common Theme: understand deterministic version first!

5. One On-Line Model vs. Its Stochastic Analog • Processing times not known in advance, but scheduler learns them by processing each job to completion • Traditional stochastic models exactly like this – BUT –adversary not controlling the processing time (even as the job is being processed) instead each job’s processing time is drawn independently according to a pre-specified probability distribution • Benchmark for competitive analysis: best off-line solution • Benchmark for stochastic analysis: optimum expected value for any non-anticipatory policy • The KEY for BOTH: GET GOOD LOWER BOUNDS

6. Example 1 Using LP to get near-optimal solutions for 1| rj | E[  wj Cj ] [Uetz; Möhring, Schulz, Uetz]

7. Start with Deterministic Single-Machine Model • Good polyhedral description for model without release dates [Queyranne; Wolsey] minimize  wj Cj subject to j2A pj Cj¸ (j2A pj )2/2 + (j2A pj2 )/2 for each A µ N • View left-hand side as minimizing  wjCj with wj = pj (and so all ratios in Smith’s rule are equal)

8. Now Extend to Identical Parallel Machines • Add constraints machine by machine to get the following valid inequalities for each feasible schedule: j2A pj Cj¸ (j2A pj )2/2m + (j2A pj2 )/2 for each A µ N • To add release dates, now just add Cj¸ rj + pj for each j 2 N

9. Valid Inequalities for P|rj |E[ wj Cj ] Let Cj be output of non-anticipatory policy: j2A E[pj]E[Cj]¸ (j2A E[pj])2/2m + (j2A E[pj]2)/2 – (j2A Var[pj])(m-1)/2m for each A µ N. Proof: Take any realization pj & let Sj denote start times. Where does non-anticipatory come in? pj and Sj are independent r.v.’s

10. Apply deterministic inequality to realization: j2A pj(Sj + pj )¸ (j2A pj )2/2m + (j2A pj2 )/2 for each A µ N • Take expectations over given distribution: j2A E[pj]E[Sj]¸ (i,j2A E[pi]E[pj])/2m + (1/2 +1/2m -1)(j2A E[pj2]) = (i,j2A E[pi]E[pj])/2m + [(m-1)/2m](j2A Var[pj]-E[pj]2 ) • Using that E[Cj] = E[Sj] + E[pj], we get desired j2A E[pj]E[Cj]¸ (j2A E[pj])2/2m + (j2A E[pj]2)/2 – (j2A Var[pj])(m-1)/2m for each A µ N.

11. Making Constraints Just Like Deterministic Ones • We start with j2 A E[pj]E[Cj]¸ (j2 A E[pj])2/2m + (j 2 A E[pj]2)/2 – (j2 A Var[pj])(m-1)/2m for each A µ N But if we let ¢ be upper bound on Var[pj ].5/E[pj] can rewrite as j2 A E[pj]E[Cj]¸ (j2 A E[pj])2/2m +®(j 2 A E[pj]2)/2 where ® = ( ¢ (1-m) +m )/2m

12. Easy to Adapt Release Date Constraints • From Cj¸ rj + pj for each j 2 N • Get E[Cj ]¸ rj + E[pj] for each j 2 N

13. A Simple LP-based Policy • Solve the LP, viewing E[Cj] as the variables – requires knowing both E[pj] and Var[pj] for the input distribution !Cj Blue is LP optimal value (lower bound) • Sort so that C1· C2· … · Cn • List-based scheduling – wait until next job on list is released and schedule on first available machine

14. Analyzing Performance Guarantee • Goal: show that Cj(¼) values computed by policy ¼ satisfy E[Cj(¼)] ·½Cj • Fix realization pj and consider job j – let J={1,2,…,j} After time R = maxk 2 J rk there can be no more idle time now, and there must exist machine with at most average load on which to process job j Cj(¼) · R + (1/m) k2J pk + pj ·Cj +(1/m) k2J pk + pj Take expectations (and use that Cj¸ E[pj] ) E[Cj(¼)] · 2 Cj+ (1/m) k2J E[pk]

15. Analysis (continued) • From before: E[Cj (¼)] · 2 Cj+ (1/m) k2J E[pk] so now use LP to bound last term j2 A E[pj]E[Cj]¸ (j2 A E[pj])2/2m +®(j 2 A E[pj]2)/2 where ® = (¢ (1-m) +m )/2m First Case: ® > 0 (small ¢ ) with A=J={1,2,…,n} (k2 J E[pk])2/2m ·k2 J E[pk] Ck·Cjk2 J E[pk] Cancelling common factor implies that (1/m) k2J E[pk] · 2 Cj Second case: (1/m) k2J E[pk] · 2(1-®) Cj

16. Theorem [Möhring, Schulz, Uetz] The LP-based policy ¼ has the guarantee, for each job j that E[Cj (¼)] · (3 + max{1,¢(m-1)/m}) Cj and thereby is a 3 + max{1,¢(m-1)/m}–approximation algorithm for P|rj|E[ wj Cj]

17. Example 2 Using the m-fast single-machine relaxation for P|pmtn, rj|E[ wjCj ] [Megow & Vredeveld]

18. The Simplest Preemptive Models 1| rj , pmtn | j Cj - Shortest Remaining Processing Time is optimal (SRPT) 1| rj , pmtn | j wj Cj Ratio rule (remaining processing time to weight) is a good approximation algorithm [Schulz & Skutella; Goemans, Wein, &Williamson] 1| pmtn |E[ wj Cj ] is solved by the Gittins Index Policy [Sevcik; Gittens; Weiss]

19. A Useful Lower Bound for  wj Cj • Consider 1 machine & machine runs at speed s • With s constant, used to bound on-line algorithms for single-machine models [Kalyanasundaram & Pruhs; Phillips, Stein, Torng, Wein] • With s=m, used to bound performance for identical parallel machine models [Chekuri, Motwani, Natarajan, & Stein] To see this, take optimal m-machine schedule, and “slice and dice” each interval without preemptions to yield single-machine schedule of no greater obj. • Still holds for optimal policy for stochastic model with somewhat more subtle proof

20. What is the Gittins Index Policy? • Consider deterministic models as a special case • Even with unit weights – Shortest Remaining Processing Time (SRPT) is optimal • Analogous Ratio rule is 2-approximation algorithm with general weight • Why? Get most completed weight per unit effort spent processing jobs

21. What is the Gittins Index Policy? 1| pmtn | E[ wj Cj ] Consider a job j that has not been processed yet Suppose we process j until q time units pass, or job j completes, whichever comes first Expected contribution to obj. fnc. = wjPr[ pj· q ] Expected cost of processing = E[min{pj,q}] Index(j) = maxq wjPr[ pj· q ]/E[min{pj,q}] Q(j) = argmaxq wjPr[ pj· q ]/E[min{pj,q}] Policy = always process job with biggest Index

22. Gittins Index Policy – More Details Once job j has biggest Index, remains job with biggest Index until first quantum Q(j) done For a job j already partially processed t time units, compute Index by considering conditional processing time distribution condit’nd that pj> t No longer optimal when there exist release dates (not surprising – deterministic setting is special case) Need complete specification of distribution Same schedule produced for machine at speed m

23. A Simple LB-Based Policy (LB=LowerBound) For P|rj|E[ wjCj ] : • At each point in time, compute m jobs with highest Index value (w.r.t. conditional remaining processing time distribution) • Run those jobs, or as many still exist (quantum argument implies that schedule stays the same until first quantum completed)

24. Analysis of Performance Guarantee

25. Analysis of Performance Guarantee • Not today – too many miles to go before I rest • But can use good characterization of Gittens optimum for single-machine input to bound policy cost

26. Theorem [ Megow & Vredeveld ] The Gittens Index Policy is a 2-approximation algorithm for P | pmtn | E[  wj Cj ]. In fact, can add release dates and get the same bound And furthermore, policy does not require knowledge of a job until it is released So performance guarantee is a mixture of competitive analysis (relative to knowing existence of jobs at time 0) and stochastic analysis (guarantee relative to what can be attained by non-anticipatory optimal policy)

27. What is the Right Performance Guarantee for Stochastic Analysis? • We bound the expected objective function value obtained by a constant times the optimal policy’s objective function value • But the inputs are random – some we may do badly, and some we may do fine • In particular, if the optimal value is “unexpectedly small” we can “afford” to produce a horrible schedule – Is the just? • [Coffman & Gilbert; Scharbrodt, Schickenger, & Steger; Souza & Steger] bound worst-case value of expected ratio of policy to optimum

28. Example 3 • Get rid of independence assumption for processing times of jobs 2-stage with recourse model of max-weight ontime set with release dates [S & Sozio] [Dye, Tomasgard, & Stougie]

29. Two-Stage Recourse Model Given : Probability distribution over inputs. Stage I : Make some advance decisions – plan ahead or hedge against uncertainty. Observe the actual input scenario. Stage II : Take recourse. Can augment earlier solution paying a recourse cost. Choose stage I decisions to minimize (stage I cost) + (expected stage II recourse cost).

30. A 2-Stage Variant of 1| rj |  wj Uj • Will consider maximization version – or equivalently – admission control variant of max-weight on-time set • Probabilistic content is not processing time, but whether job exists or not • Potentially 2n realizations – how do we specify distribution? Three Possible Approaches • Only a polynomial number of subsets A occur with probability pA • Existence of jobs are independent random events • Black box model – can draw independent samples to learn the distribution

31. Maximum-weight on-time set Jobs N = {1,2,…,n}- job j has set of allowed time intervals Sj = {[s1j,e1j),…,[skj,ekj)} with corresponding weights wij Deterministic problem: Pick a maximum-weight collection of intervals 1 per job and at each time Height = Weight Time

32. Maximum-weight on-time set Jobs N = {1,2,…,n}- job j has set of allowed time intervals Sj = {[s1j,e1j),…,[skj,ekj)} with corresponding weights wij Deterministic problem: Pick a maximum-weight collection of intervals 1 per job and at each time Height = Weight Optimal selection is shaded Time

33. Maximum-weight on-time set Jobs N = {1,2,…,n}- job j has set of allowed time intervals Sj = {[s1j,e1j),…,[skj,ekj)} with corresponding weights wij Deterministic problem: Pick a maximum-weight collection of intervals 1 per job and at each time Linear Programming Relaxation Let Tt be the set of intervals (for all jobs) containing time t : (i,j) xij : indicates whether [sij,eij) selected for job j Maximize i,j wij xij Subject to i xij 1, for each j=1,…,n (i,j)  Tt xij1 for each t xij 0 for each i,j Theorem [Bar-Noy, Bar-Yehuda, Freund, Naor, & Schieber] Primal-dual 2-approximation algorithm for max-weight schedule

34. 2-Stage Stochastic Variant Scenario A  N of active jobs occurs with probability pA Stage I: Choose setD  Nof jobs todeferto subcontractor and receive small weightj for each j  D Stage II: Given realized scenario A, make selectionTAwhere(i,j)  TA j  A-Dand has weightWij Goal: Maximize the total expected weight scheduled (where expectation is with respect to active subset probabilities)

35. A Primal-Dual Theorem Theorem: [S & Sozio] We can (adapt the2-approximation algorithm for deterministic setting to) obtain a 2-approximation algorithmfor stochastic maximum-weight on-time scheduling. We focus first on the polynomial-scenario model

36. Let Tt be the set of intervals (for all jobs) containing time t • Minimize j uj + t vt • Subject to • uj + t: (i,j)  Tt vt wij for each (i,j) • uj, vt 0 • The primal-dual algorithm has two phases: • first it constucts a feasible dual solution, while building a stack of possible pairs (i,j) to be selected; • next it pops the stack, selecting any pair that doesn’t conflict with those already selected; • amortization shows dual cost is at most twice the value of the primal. Dual Linear Program

37. Linear Programming Relaxation for 2-Stage Problem Let Tt be the set of intervals (for all jobs) containing time t xj : indicates whether job j is deferred in stage I yij(S): indicates whether [sij,eij) selected for job j in stage II for scenario S Maximize jjxj + i,j,S pS Wij yij(S) Subject to xj + i yij (S) 1, for each j,S (i,j)  Tt yij(S) 1, for each t,S xj , yij (S)  0 for each i,j,S DUAL Minimize j,S uj (S) + t,S vt (S) Subject to S uj(S)j for each j uj(S) + t: (i,j)  Tt vt(S)  p(S) Wij for each (i,j), S uj (S), vt(S)  0

38. A Simple 2-Stage Algorithm For each scenario A  N with probability pA >0 run the deterministic algorithm with job set A where weight of job j for [sij,eij) ispA Wij let uj (A) denote the dual values constucted by the algorithm Stage I: Let D be the set of jobs j for which j > A uj(A) Stage II: Given realized scenario A, recompute first phase of algorithm (to get duals) but in second phase never select (i,j) for j  D

39. Main Idea of Analysis What is 2-stage dual? Block-structured by scenario A with additional linking constraints: A1 Each block is just like determ. problem A2 Am A uj(S) j j So we can adapt the scenario-by-scenario constuction as building a feasible dual solution for the 2-stage linear relaxation

40. What about black box model? Just use sampling to estimate the deferral rule! - use M samples For each sampled scenario A  N run deterministic algorithm with job set A where weight of job j for [sij,eij) isWij to obtain dual values uj(A) – let Ak be kth sample Stage I: Let D be the set of jobs j for which (1+) j > (1/M) k uj(Ak) Stage II: Given realized scenario A, compute TA and then recompute first phase of algorithm (to get duals) but in second phase never select (i,j) for j  D Number of samples needed is polynomial in n, /, and  = maxj Wj/j Similar to “sample average approximation” results of [Swamy & S, Shapiro & Nemirovski, and Charikar, Chekuri, & Pál]

41. Example 4 The a priori TSP – a stochastic sequencing model (again where probabilistic content is existence of a task) [ S & Talwar ]

42. A priori optimization (no recourse) Given: Probability distribution over inputs. In advance: Compute master plan. Observe the actual input scenario. In real time: Adapt master plan to scenario. Compute master plan to minimize expected real time cost.

43. The Traveling Salesman Problem (TSP) Given input points, compute tour  to minimize total length c()

44. The Traveling Salesman Problem (TSP) Given input points, compute tour  to minimize total length c()

45. The A Priori TSP Given input points N and a distribution  of active sets A 2 2N Need to specify the probability that a given set A is active

46. The A Priori TSP Active Nodes Given input points N and a distribution  of active sets A 2 2N Need to specify the probability that a given set A is active

47. The A Priori TSP Active nodes A Given input points N and a distribution  of active sets A 2 2N, compute master tour to minimize expected length of the tour  shortcut to serve only A

48. The A Priori TSP Active nodes A Shortcut tour on A Given input points N and a distribution  of active sets A 2 2N, compute master tour to minimize expected length of the tour  shortcut to serve only A

49. The A Priori TSP Active nodes A Shortcut tour on A Given input points N and a distribution  of active sets A 2 2N, compute tour  to minimize expected length EA [c( A )], where A is the tour  shortcut to serve only A

50. The A Priori TSP Active nodes A Optimal tour on A Given input points N and a distribution  of active sets A 2 2N, compute tour  to minimize expected length EA [c( A )], where A is the tour  shortcut to serve only A