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Coordination Mechanisms for Unrelated Machine Scheduling Yossi Azar joint work with Kamal Jain Vahab Mirrokni

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Price on Anarchy [KP, RT] Selfish users User goal: minimize its cost Nash Equilibrium (NE) System goal (e.g. Social welfare) The worst ratio of NE cost to OPT cost

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Price of Anarchy Concept Not algorithmic Only analysis What to do if PoA is large How to influence the system

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Possible Solutions Change the system (add tolls, payments) Stackelberg strategy = control some users Disadvantages: changing the settings, global knowledge Challenge: influence within the same setting and locally (distributed)

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Coordination Mechanism [CKN] Mechanism: local policy (algorithm) that assigns a cost for each strategy of the user Advantages: local, same type of cost Goal: achieving good NE Example: scheduling jobs on machines

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Unrelated Machine Scheduling m unrelated machines n jobs – each owned by different user p(i,j) - processing time of job i on machine j System goal: minimize completion time User goal: minimize its own completion time m unrelated machines n jobs – each owned by different user p(i,j) - processing time of job i on machine j System goal: minimize completion time User goal: minimize its own completion time

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Unrelated Machines Scheduling

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Coordination Mechanism for Scheduling Policy for each machine (algorithm) which decides how to schedule jobs assigned to it Each Policy induces NE on jobs

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Local Scheduling Policies

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Type of Policies Local policy – depends on jobs assigned to machine Strongly local policy - depends only on processing time of jobs on that machine Ordering Policy = IIA (independence of irrelevant alternative)

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Challenge Design policies that results in good NE (i.e. low PoA)

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PoA of Longest First Results in poor NE The PoA is unbounded even for 2 machines The optimum completion time is low The completion time of NE is large

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Unrelated Machines Scheduling

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Equilibrium for Longest First

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Previous Results Identical Machines: constant [CKN] Related: constant, log m [CV,ILMS] Restricted assignment: log m [ILMS] Unrelated Machines: m (IK,DJ,ILMS)

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Main Results Negative Results (strongly local): PoA of any strongly local policy-at least m/2 In particular, PoA of Shortest-First is of order m Resolve an open question from 1977 (Alg D by Ibarra and Kim)

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Main Results Positive Results (local): Local ordering policy with PoA of O(log m) Any local ordering policy – at least log m Pure Nash + Convergence O(log^2 m) More results on convergence …

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Lower Bound for Strongly Local Policy We start with Shortest-First Extend it to arbitrary strongly local IIA policy Shortest-First is interesting by its own

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Shortest-First Approx factor known to be at most m NE can be computed by shortest-first greedy algorithm (Alg D by Ibarra and Kim) An open question from 1977 We show it is at least m/2

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Idea of the Proof m types of jobs Type j can be scheduled on machines j & j+1 Processing time of type j on machine j is low and on machine j+1 is high (ratio is j) All jobs on machine j have almost the same processing time

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Example for Shortest-First

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Idea of the Proof OPT assign all jobs of type j to machine j Number of jobs is chosen such that OPT has the same completion time for all machines

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Optimal Assignment

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Idea of the Proof In NE about half jobs of type j are on machine j and half on machine j+1 Completion time of NE grows linearly in m

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Equilibrium for Shortest-First

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Extend to Arbitrary Strongly Local Structure is similar to lower bound for Shortest-First Arbitrary ordering function is given for each machine Indices of jobs are chosen to behave similar to the above example

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Efficiency Based Algorithm Order jobs on each machine by their efficiency Efficiency of job on machine is: The ratio between jobs best processing time to its processing time on this machine PoA of algorithm is O(log m)

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Equilibrium Improves

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Efficiency Based Algorithm Unfortunately – pure NE may not exist Iterative improvement may cycle Modified algorithm guarantees convergence and pure NE with PoA of O(log^2 m)

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Modified Algorithm Each machine simulate log m submachines (by round robin) Submachine k of machine j handles jobs on efficiency between 2^{-k} and 2^{-k+1} Jobs are ordered on submachine by Shortest-First PoA of algorithm is O(log^2 m)

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Summary Coordination Mechanism: Influence on the quality of the equilibrium Unrelated Machines: m – lower bound Shortest-First is at least m Local order by efficiency O(log m) – optimal Pure + Convergence O(log^2 m)

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Discussion and Open Problems Non ordering strategies – get below log m Extend to network routing Show more effective usage of coordination mechanism

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