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On Scheduling Mechanisms : Theory, Practice and Pricing

On Scheduling Mechanisms : Theory, Practice and Pricing. Ahuva Mu’alem SISL, Caltech. Motivation. Mechanisms ≈ auctions & reverse- auctions ≈ optimization problems with strategic constraints. Scheduling Problem. n jobs to be assigned to m machines

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On Scheduling Mechanisms : Theory, Practice and Pricing

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  1. On Scheduling Mechanisms: Theory, Practice and Pricing AhuvaMu’alem SISL, Caltech

  2. Motivation • Mechanisms ≈ auctions & reverse- auctions ≈ optimization problems with strategic constraints

  3. Scheduling Problem • njobs to be assigned to mmachines • tij = time required to process job j on machine i • Goal: Minimize the maximum load (“makespan”) • It’s a well-studied NP-hard problem with [1.5, 2] approximability lower and upper bounds [Lenstra, Shmoys, Tardos’87]

  4. Example: 2 jobs, 3 machines • The optimal allocation has a makespan of 1 • Any other allocation has makespan > 1 • Machine m2 is related to m3 but not to m1 • rank > 1 is called “multi-dimensional”

  5. The Mechanism Design Problem • njobs to be assigned to mstrategicmachines • Machine i has a privatecostci(j) = tij • Goal: Design a scheduling algorithm ALG and a compensation function p (payment) such that the mechanism M(ALG, p) minimizes the makespan in a truthful manner (reporting its true private cost is a dominant strategy for any strategic machine, assuming quasi-linearity)

  6. In their seminal paper [Nisan, Ronen ’99] asked: How well this goal can be approximated in a TRUTH-TELLING manner? • Thesingle-dimensional case is solved! A deterministic truthful (1+ε)-approximation mechanism exists in time polynomial(m,n), if all machines are related [Archer, Tardos ’01], [Auletta et al. ’04], [Andelman et al. ’05 + ’07], [Kovacs ’05 + ’07], [Dhangwatnotai, Dobzinski, Dughmi, Roughgarden ‘08], [Christodoulou, Kovacs ’10]

  7. The Multi-Dimensional Case

  8. Deterministic Truthful Mechanism • Example: m1 gets 3 jobs, and is “truthfully” paid 3, resulting in a makespan of 3-3ε; the optimal is 1 • Can we do better w.r.tmakespan? • Job-by-Job Mechanism [NR]: Assign each job to the fastest machine and pay the 2nd cheapest cost

  9. Specifically, the OPTIMAL algorithm w.r.tmakespan cannot be truthfully implementable. This justify our focus on APPROXIMATION algorithms

  10. Truthful Randomized Mechanisms • Definition: A truthfulrandomized mechanism is a probability distribution DM over truthfuldeterministic mechanisms (“with the same DM for every declared cost”) • Examples: (1) “Random Dictator”; (2) Run the Job-by-Job mechanism on 2 machines selected uniformly at random

  11. Randomized Lower Bounds • Thm [M, Schapira]: Any truthful randomized mechanism for minimizing the makespan cannot achieve approximation ratio better than 2-1/m. The same holds for truthfulness in expectation(using a different proof technique). • Remark: very few GT-LBs are known for randomized truthful mechanisms

  12. Proof Idea Yao’s Principle: Weak-Monotonicity: Theorem [BCRMNS ‘06]: If M(ALG, p) is a truthful mechanism then for every costs ci,di, c-i it holds that ci(Si) + di(Ti) ≤ di(Si) + ci (Ti) where ALG(ci, c-i) = Si and ALG(di, c-i) = Ti • Find a probability distribution DCover machine costson which anytruthfuldeterministic mechanism fails to provide the expected approximation of 2-1/m w.r.t makespan

  13. Proof Idea Yao’s Principle: Weak-Monotonicity: Thm[Roberts79],[Rochet87]: If M(ALG, p) is a truthful mechanism, then for any costs ci,di, c-i it holds that ci(Ci) + di(Di) ≤ di(Ci) + ci(Di) where the subset of jobs Ci, Di are defined by ALG(ci, c-i) = Ci and ALG(di, c-i) = Di • Find a probability distribution over inputs on which any truthful deterministic mechanism fails to provide the expected approximation ratio of 2-1/m w.r.t makespan

  14. Approximations ALG(c1, c2) ALG(d1, c2) ALG(c1, d2)

  15. TruthfulApproximations ALG(c1, c2) ALG(d1, c2) ALG(c1, d2)

  16. Randomized Lower Bound The probability of each input is: p1 = ε, p2 = p3 = (1-ε)/2. Case 1: If the deterministic mechanism on the first input has a sub-optimal makespan the expected ratio then is at least p1·(99 /ε)/8 > 3/2 Case 2: Otherwise, suppose wlog it allocates j1 to m1, and j2, j3 to m2, the best it can do on the third input is a makepan of 8 (without violation of weak-monotonicity), the expected ratio then is at least: p2 · 8 / (4+2ε) + p3 · 1 = 3/2 - ε’

  17. Randomized Lower Bound The probability of each input is: p1 = ε, p2 = p3 = (1-ε)/2. Case 1: If the deterministic mechanism on the first input has a sub-optimalmakespan, the expected ratio then is at least p1· (99 /ε) / 8 > 3/2 Case 2: Otherwise, suppose wlog it allocates j3to m2, the best it can do on the third input is a makepan of 8 (without violation of weak-monotonicity), the expected ratio then is at least: p2 · 8 / (4+2ε) + p3 · 1 = 3/2 - ε’

  18. Randomized Lower Bound The probability of each input is: p1 = ε, p2 = p3 = (1-ε)/2. Case 1: If the deterministic mechanism on the first input has a sub-optimalmakespan, the expected ratio then is at least p1· (99 /ε) / 8 > 3/2 Case 2: Otherwise, suppose wlog it assigns j3to m2, the best it can do on the third input is a makepan of 8 (without violation of weak-monotonicity), the expected ratio then is at least: p2 · 8 / (4+2ε) + p3 · 1 = 3/2 - ε’

  19. Envy-Free Design • M(ALG, p) is an envy-free design if p(Si) - ci(Si) ≥ p(Sk) - ci(Sk) for every 1≤ i, k ≤m, whereALG(c) = (S1, S2, …, Sm) [M’09] How well the makespan can be approximated in an ENVY-FREE manner (“no agent is willing to exchange his allocated bundle and payment with any other agent”)?

  20. Envy-Free Design • M(ALG, p) is an envy-free design if p(Si) - ci(Si) ≥ p(Sk) - ci(Sk) for every 1≤ i, k ≤m, whereALG(c) = (S1, S2, …, Sm) • Motivation: • BI-CRITERIA optimizations with INDIVIDUAL-LEVEL GUARANTEE • Envy-freeness can lead to dominant strategy mechanisms (e.g., Ascending Auctions with Budgets [Aggarwal et al.‘09]) • Study Algorithms & Pricing for multi-parameter problems

  21. Commercial Clouds Simulations on Real-Data: measure “average-case” scenarios, also allow us to study several aspects simultaneously

  22. Mechanism Design Challenges • Provider’s Goals: Revenue and Quality of Service • vs. • Users’ Strategic Behavior • On-LineSetting: jobs/tasks arrive over time • Uncertainties about run time • Our Approach [Shudler, Amar, Barak, M. ‘10]: • Simulation-based analysis performed on real data taken from The Parallel Workload Archive @HUJI

  23. Setting • Homogeneous Cluster with identical machines • Each user submits a single job • The type of job j is denoted by: ( rj , tj , wj ) • rj > 0 is therelease time (“arrival time”) • tj > 0 is the running time (unknown to the user) • wj> 0 is thevalue per unit time of delay

  24. Setting (cont.) • The utility of job j is uj = -wj Fj - pj • Fj is the flow time: duration from arrival to completion • pj > 0 is the payment of job j • uj < 0, [Heydenreich, Muller and Uetz ’06]. • Remark 1: tj and wj are independent • Remark 2: to generate wj we used a bimodal distribution

  25. The SRG Model • Honest Arrivals and Runtimes • Big Conservative Group: • 90% of the users always declare wj [0.9 wj , wj ] uniformly at random • Small Aggressive Group: • 10% of the users declare wj [0.1 wj , wj ] uniformly at random. Aggressive users respond to incentives

  26. Stability Analysis We formulate a simple one-shot game to model the dynamic interaction between the provider and an aggregate consumer playing on behalf of the aggressive users We then look for a Nash-equilibrium in this restricted game

  27. Algorithms • HB Algorithm: Upon any job arrival or termination, preempt all running jobs and run the waiting jobs with the highest declared wj • HBNP Algorithm: Upon any job termination run the waiting job with the highest declared wj . • WSPT Algorithm: Upon any job termination run the waiting job with the highest declared wj / tj . • Remark: WSPT has informational advantage by knowing tj.

  28. Aggregate user’s payoff is the summation of all aggressive user utilities: ∑uj • The “β%” Strategy means that wj  [(1- β) wj , wj ] uniformly at random. • Every line has a single best response ! (marked above in red) • The k-th price best responses are more “truthful” !

  29. The Provider’s payoff is a function of the Total Revenue and the QoS (total weighted flow time): • REV* = ∑ pj / ∑ wj Fj • REV* nicely behaves: Left column always has the highest values and right column has the smallest values. • 1st Price: HB is the best w.r.t. QoS, NPHB is the worst.

  30. Nash-Equilibrium for 1st price is [HB, 50%] with a near-optimal REV* (0.976 ≈ 1.000). • Nash-Equilibrium for k-th price (ignoring the WSPT) is [NPHB, 25%]: using a non-socially optimal scheduler increases REV* (prices increase when many high value jobs are delayed in the waiting queue. The aggregate consumer is almost truthful in the NE ).

  31. Conclusions (Empirical Part) • We introduced the SRG Model: a simple behavioral model to study scenarios with inherent uncertainties. • We modeled the dynamic on-line interaction between the provider and consumers as a one shot game and showed the existence of (arguably good) unique ‘pure’ symmetric Nash Equilibrium. • Future Work: • Non-linear value and utility models. • Strategic impact of budgets (runtime uncertainty causes unpredicted payments). • Competition among providers in a more direct manner.

  32. Thank You

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