Coordination Mechanisms for Unrelated Machine Scheduling

Yossi Azar joint work with Kamal Jain Vahab Mirrokni

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

Price of Anarchy Concept

• Not algorithmic • Only analysis • What to do if PoA is large • How to influence the system

Possible Solutions

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

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

Unrelated Machine Scheduling

Unrelated Machines Scheduling

?

A B

Machine A Machine B ?

B A

Coordination Mechanism for Scheduling

Policy

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

Local Scheduling Policies

A A Shortest-First Policy A A Longest-First Policy B B

Type of Policies

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

Challenge

Design policies that results in

good

NE (i.e. low PoA)

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

Unrelated Machines Scheduling

?

A B

Machine A Machine B ?

B A

Equilibrium for Longest First

A B A B

Previous Results

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

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)

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 …

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

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

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

Example for Shortest-First

?

?

A B C ?

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

Optimal Assignment

A B C

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

Equilibrium for Shortest-First

?

?

A B C ?

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

Efficiency Based Algorithm

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

Equilibrium Improves

?

?

A B C ?

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)

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)

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)

Discussion and Open Problems

• Non ordering strategies – get below log m • Extend to network routing • Show more effective usage of coordination mechanism