Job Scheduling

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Job Scheduling

• Lecture 19 March 19

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Job Scheduling Unrelated Multiple Machines

• There are n jobs, each job has
• a processing time p(i,j) (the time to finish
  this job j on machine i)

There are m machine available.
Task to scheduling the jobs -To minimize the
completion time of all jobs (the makespan)
NP-hard to approximate within 1.5 times of the
optimal solution.
Well design a 2-approximation algorithm for this
problem.
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Why Unrelated?
For example, different processors have different
specialties.
Computational jobs, display images, etc
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Job Scheduling Unrelated Multiple Machines

• There are n jobs, each job has
• a processing time p(i,j) (the time to finish
  this job j on machine i)

There are m machine available.
Task to scheduling the jobs -To minimize the
completion time of all jobs (the makespan)
Approach Linear Programming.
How to formulate this problem into linear program?
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Linear Programming Relaxation
whether job j is scheduled in machine i
Each job is scheduled in one machine.
for each job j
Each machine can finish its jobs by time T
for each machine i
Relaxation
for each job j, machine i
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How good is this relaxation?
Example
for each job j
One job of processing time K for each machine
for each machine i
Optimal solution K.
Optimal fraction solution K/m.
for each job j, machine i
The LP lower bound could be very bad.
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How good is the relaxation?
Example
for each job j
One job of processing time K for each machine
for each machine i
Optimal solution K.
Optimal fraction solution K/m.
for each job j, machine i
Problem of the linear program relaxation an
optimal solution T could be even smaller than the
processing time of a job!
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How to tackle this problem?
Problem of the linear program relaxation an
optimal solution T could be even smaller than the
processing time of a job!
Ideally, we could write the following constraint
but this is not a linear constraint
Idea?
To enforce this constraint by preprocessing!
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Preprocessing
Fix T. Consider the decision problem instead of
an optimization problem
Call the resulting linear program LP(T). Note
that different T have different linear programs.
This is known as parametric pruning.
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Decision Problem
Fix T
Let S(T) be the set of jobs with p(i,j) lt T.
for each job j
for each machine i
for each job j, machine i
Use binary search to find the minimum T such
that this LP is feasible.
We will use T as the lower bound on the value of
an optimal solution, clearly T lt OPT, since
LP(OPT) is feasible.
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Basic Solution
for each job j
for each machine i
for each job j, machine i
What can we say about a vertex solution of this
LP?
Basic solution unique solution of n linearly
independent tight inequalities, where n is the
number of variables.
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Basic Solution
for each job j
for each machine i
for each job j, machine i
A tight inequality of the last type corresponds
to a variable of zero value.
There are at most nm inequalities of the first
two types, and hence there are at most nm
nonzero variables.
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Basic Solution
Say a job is integral if it is assigned entirely
to one machine otherwise a job is fractional.
Each fractional job is assigned to at least two
machines. Let p be the number of integral
jobs, and q be the number of fractional jobs.
There are at most nm nonzero variables.

• p q n
• p 2q lt n m

• p gt n m
• q lt m

There are at most m fractional jobs.
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Integral Jobs
How to handle integral jobs?
Just follow the optimal fractional solution.
And so we can schedule all the integral jobs in
time at most T lt OPT, as this schedule (on
integral jobs) is just a subset of the fractional
solution.
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Fractional Jobs
Observation Suppose there are m machines and at
most m jobs. If we can assign all jobs to the m
machines so that each machine is assigned at most
1 job, then the completion time (makespan) is at
most T lt OPT.
There are at most m fractional jobs.
If we could find such a matching, then we use
this matching to schedule all the fractional jobs
in time at most T lt OPT.
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Approximation Algorithm
Goal to design a 2-approximation algorithm for
this problem

• Do preprocessing (parametric pruning) and find a
  smallest T so that LP(T) is feasible.
• Find a vertex (basic) solution, say x, to LP(T).
• Assign all integral jobs to machines as in x.
• Match the fractional jobs to the machines so that
  each machine is assigned at most one job.

Proof (assuming a matching exists) Schedule all
integral jobs in time T, Schedule all fractional
jobs in time T, Schedule all jobs in time 2T lt
2OPT.
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Bipartite Matching
Task Match the fractional jobs to the machines
so that each machine is assigned at most one job.
Create a vertex for each job j, and create a
vertex for each machine i, add an edge between
machine i and job j if 0 lt x(i,j) lt 1.
Now, the problem is to find a matching so that
every job is matched.
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Bipartite Matching
job machine
Assume the graph is connected.
There are at most nm nonzero variables.
n m vertices, n m edges, at most one cycle.
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Bipartite Matching
Leaves must be machines, since each fractional
job is adjacent to two machines.
Match a leaf machine with its adjacent job, then
remove these vertices and repeat.
n m vertices, n m edges, at most one cycle.
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Bipartite Matching
Match a leaf machine with its adjacent job, then
remove these vertices and repeat.
Eventually a cycle is left, and we can find a
perfect matching.
n m vertices, n m edges, at most one cycle.
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Bipartite Matching
If the graph is not connected, we apply the same
argument to each connected component.
Prove (1) each component has at most nm
edges. (2) each component has a
matching.
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Bad Examples
m machines
m2 m 1 jobs 1 job of processing time m on
all machines remaining jobs have processing time
1 on all machines
Optimal solution the large job on one machine, m
small jobs on the remaining m-1
machines, makespan m
LP vertex solution 1/m of the first job and m-1
other jobs to each machine. Our
rounding procedure will produce a schedule of
makespan 2m-1.
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General Assignment

• There are n jobs, each job has
• a processing time p(i,j) (the time to finish
  this job j on machine i)
• a processing cost c(i,j) (the cost to finish
  this job j on machine i)

There are m machine available.
Task to scheduling the jobs -To minimize the
total cost of the assignment -Satisfying time
constraint T(i) for each machine
Theorem. Let OPT be the optimal cost to satisfy
all constraints. There is a polynomial time
algorithm which finds an assignment with cost at
most OPT and the constraint is violated at most
twice.
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Linear Programming Relaxation
for each job j
for each machine i
for each job j, machine i
Pruning Delete every variable x(i,j) with p(i,j)
gt Ti
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Iterative Relaxation
Iterative General Assignment Algorithm (Basic
solution) Compute a basic optimal solution of
the LP. Delete every variable x(i,j) with
x(i,j)0 (Assigning a job) If there is a
variable with x(i,j)1, assign job j to
machine i, and set Ti Ti p(i,j). (Relaxing a
constraint) If there is a machine i with only
one job, or there is a machine with two jobs j1
and j2 and x(i,j1) x(i,j2) gt 1, remove the
time constraint for machine i.
repeat
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Performance Guarantee
Lemma. Suppose the algorithm terminates.
Then the total cost is at most LP,
and each time constraint is violated at most
twice.

• Deleting a job of value 0 does not change
  anything.
• Assigning a job of value 1 keeps the total cost
  and the constraints satisfied.
• Relaxing a machine i with only one job can add
  at most Ti to machine i.
• Relaxing a machine i with two jobs j1 and j2
  with x(i,j1) x(i,j2) gt 1.
• In the worse case, both j1 and j2 are assigned
  to machine i (in future).
• Then x(i,j1)p(i,j1) x(i,j2)p(i,j2) Ti gt
  p(i,j1) p(i,j2),
• and so the constraint is violated by at most
  Ti.

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Counting Argument
Lemma. If there is no variable with value 0 or
1, then the relaxation step applies.
job machine
n jobs
m machines

• Each job has degree at least 2 (otherwise there
  is a job with value 1).
• Each machine has degree at least 2 (otherwise
  the relaxation step applies).
• So there are at least nm edges and thus nm
  nonzero variables.
• There are n jobs and m machines, so there are nm
  constraints,
• and so in a basic solution, there are at most
  nm nonzero variables.

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Counting Argument
Lemma. If there is no variable with value 0 or
1, then the relaxation step applies.
job machine
n jobs
0.3
0.7
0.7
0.2
0.3
0.8
m machines

• So there are exactly mn edges, and so is a
  disjoint union of cycles.
• So each machine has degree exactly 2.
• Each job has total value 1.
• So there exists a machine with total value at
  least 1.

The relaxation step applies!
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Remarks

• There are many more scheduling problems in the
  literature.
• Iterative relaxation method is very useful.
• Project outline
• Meeting signup
• 5 more lectures to go.