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Stackelberg Scheduling Strategies

Stackelberg Scheduling Strategies. By Tim Roughgarden. Presented by Alex Kogan. Abstract. We consider setting of scheduling jobs on a set of shared machines with load-dependent latency functions. The system performance is measured by the total latency of the system.

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Stackelberg Scheduling Strategies

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  1. Stackelberg Scheduling Strategies By Tim Roughgarden Presented by Alex Kogan

  2. Abstract • We consider setting of scheduling jobs on a set of shared machines with load-dependent latency functions. The system performance is measured by the total latency of the system. • We assume that the users selfishly wish only to minimize the latencies of their own jobs. • In this case, the total latency is non-optimal.

  3. Abstract (cont.) • If there’s a mix of “selfishly controlled” jobs and “centrally controlled” jobs, the assignment of centrally controlled jobs will influence the subsequent actions of the selfish users. • We’re interested in assigning the centrally controlled jobs in the best possible way (that maximizes the overall system performance ).

  4. Coping with Selfishness • In many large-scale systems (like the Internet), there is no central authority controlling the allocation of shared resources. • The users act selfishly (non-cooperative game). • Results in Nash eq. => sub-optimal performance. • Given a system with a mix of centrally and selfishly controlled jobs, how can centrally controlled jobs assignment to induce “good” behavior from the non-cooperative users?

  5. Stackelberg Games • The roles of different players are asymmetric. • One player acts as a leader (according to some strategy). • All other agents (the followers) react independently and selfishly to the leader’s strategy, reaching a Nash equilibrium relative to the leader’s strategy. • The Stackelberg equilibrium is the minimum-cost equilibrium achieved by a Stackelberg strategy.

  6. The Central Questions • Given a set of m machines with load-dependent latencies and a large number of very small jobs to be scheduled, we can ask: • Among all leader strategies for a given set of machines and jobs, can we characterize and/or compute the strategy inducing the Stackelberg equilibrium - i.e., the eq. of minimal total latency? • What is the worst-case ratio between the total latency of the Stackelberg eq. and that of the optimal assignment of jobs to the machines?

  7. Results • We give a simple polynomial-time algorithm algorithm for computing a leader strategy that induces an equilibrium with total latency no more then 1/ times the optimal ( - the fraction of centrally controlled jobs). • We give an O(m2) algorithm for computing a strategy inducing total latency of at most 4/(3+) of the optimal in special case of linear latency functions.

  8. Results (cont.) • Computing the strategy inducing the Stackelberg equilibrium is NP-hard, even when the latencies are linear!

  9. The Model • Set M of m machines 1, 2, …, m • li() is the latency of machine i (continuous and non-decreasing). • (M, r) - an instance with machines M, rate r and no centrally controlled jobs. • (M, r, ) - a Stackelberg instance, where (0,1) indicates the fraction of the centrally controlled traffic.

  10. Stackelberg Strategies and Induced Equilibria • Definition: A Stackelberg strategy for the Stackelberg instance (M, r, ) is an assignment feasible for (M, r). • Definition: Let s be a strategy for Stackelberg instance (M, r, ) where machine i has latency function li, and let li~(x) = li(si+ x) for each iM. An equilibrium induced by s is an assignment t at Nash equilibrium for (M,(1-)r) w.r.t. latency functions li~.

  11. The Aloof Strategy • If x* is the optimal assignment for (M, r), put s = x*. • The minimum-cost strategy (ignoring the existence of jobs that are not centrally controlled). • Poor performance.

  12. The Scale Strategy • If x* is the optimal assignment for (M, r), put s =  x*. • The optimal assignment of the jobs, suitably scaled. • Poor performance.

  13. The LLF Strategy • Both the Aloof and the Scale strategies suffer from the same flaw: both don’t consider the selfish users behavior. • It’s reasonable for a good strategy to give priority to the machines that are least appealing to selfish users - machines with relatively high latency. • We consider the Largest Latency First strategy.

  14. The LLF Strategy (cont.) • The LLF steps: • Compute the optimal assignment x* for (M, r) • Index the machines of M so that l1(x1*)  ...  lm(xm*) • Let k m be minimal with i>kxi* r • Put si= xi* if i > k, sk = r - i>kxi*, si= 0 if i < k • A machine i is saturated by s if si= xi*. LLF saturates machines of the largest latency until there’re no centrally-controlled jobs remaining.

  15. The LLF Performance Guarantee • For arbitrary latency functions, the LLF always induces an assignment of cost no more than 1/ times that of the optimal assignment. • can be computed in polynomial time • For linear latency functions, the LLF performance guarantee is 4/(3 + ). • can be computed in O(m2)

  16. The Complexity of Computing Optimal Strategies • The LLF strategy not always provides the optimal result. • The problem of computing the optimal Stackelberg strategy is NP-hard, even for instances with linear latency functions.

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