BECO 2004

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BECO 2004

• When can one develop an FPTAS for a sequential
  decision problem?
• with apologies to Gerhard Woeginger
• James B. Orlin
• MIT
• working jointly with Mohamed Mostagir

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Fully Polynomial Time Approximation Scheme
(FPTAS)

• INPUT.
• A sequential decision problem with n stages or
  decisions. Also, a given accuracy ?.
• OUTPUT
• A solution that is guaranteed to be within ? of
  optimal.
• RUNNING TIME
• Polynomial in the size of the problem and in 1/ ?.

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Goal of this talk

• Present Branch and Dominate a generic method
  for creating FPTASes
• Starting point for this research Woeginger
  2000
• applies to all problems considered by Woeginger
• Generalizes results to multiple criteria as per
  Angel, Bampis, and Kononov 2003
• Extends to many dynamic lot-sizing and many new
  problems.

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Overview of what comes next

• Two examples
• knapsack
• a more complex machine scheduling problem
• A generalization to many other FPTAS

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Example 1. The Knapsack Problem

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The Knapsack Problem as a Decision Problem
x3 1
x2 1
x3 0
x1 1
x3 1
Decision tree.
x2 0
x3 0
Enumeration Tree
x3 1
x2 1
x3 0
x1 0
x3 1
x2 0
x3 0
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Domination

• Let x denote a decision node at stage j.
• its state is ?v, w?, where
• v c1x1 cjxj
• w a1x1 ajxj
• x is infeasible if w gt b.
• Let x have state ?v, w? at stage j
• Node x dominates node x at stage j if
• v ? v and w ? w.

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A pseudo-polynomial time algorithm

• Branch and Dominate (B D)
• Expand the enumeration tree one stage at a time.
• Eliminate any infeasible nodes
• Whenever one node dominates another, eliminate
  the dominated node. (Do this sequentially)
• The optimum corresponds to the stage n node with
  greatest value.
• Theorem. Branch and dominate is
  pseudo-polynomial for the knapsack problem.

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A simple Example

• Maximize 6x1 8x2 14x3
• Subject to 7x1 5x2 12x3 ? 100

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The Knapsack Problem as a Decision Problem
Max 6x1 8x2 14x3 s.t. 7x1
5x2 12x3 ? 100
0,0
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?-domination

• Node x ?-dominates node x at stage j if
• v ? (1- ?) v and w ? w.
• The number of undominated states at each stage is
  O(?-1 log nCmax).
• Theorem. Branch and ?-dominate with ? ?/n is
  an FPTAS for the knapsack problem. The running
  time is O(n2/?)

• Note we did not use w ? (1 ?) w because there
  is a hard constraint on knapsack weights, and we
  cannot approximate it.

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Outline of Proof of ?-optimality

• Let x (x1, x2, , xn) be the optimal solution.
• Let yj be a partial solution at stage j for j 0
  to n.
• y0 ?.
• for each j 1 to n, yj yj-1,xj or else
  yj is the solution that ?-dominates yj-1,xj
• Let xj (x1, , xj).
• Then yj j? -dominates xj for each j.
• This is the standard construction

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y4
y5
y2
y0
y1
y2 ?-dominates x2.
y4 ?-dominates w4.
y5 ?-dominates w5.
Total accumulated error at most n?.
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List Scheduling Problems

• Scheduling problems in which jobs are constrained
  to be assigned in sequential order.

5
12
6
13
15
Machine 1
14
2
1
3
10
Machine 2
8
7
4
9
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Machine 3

• Finding an optimal list schedule
• finds the optimal solution for some problems
• minimize sum of completion times on K machines.
• can be used as a heuristic for an NP-hard
  problem

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A 2-machine List Scheduling Problem
Cj completion time
Cj . is defined correctly for j 1 to n
same number of jobs on each machine
proc. time bounds on machine 1.
x ? 0, 1n
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Stage j after j jobs have been assigned
Each partial solution xk has an associated state
vector.
M1(j)
processing time on machine 1 at most bj
M2(j)
processing time on machine 2
d(j)
number of jobs on machine 1 number of jobs on
machine 2
z(j)
cumulated objective function
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On moving from Stage j to Stage j1
Stage 12
30
35
3
530
x12 1
p12 3
F11(27, 35, 2, 500, 1) ? 30, 35, 3, 530 ?
current state decision ? state at next stage
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Polynomially Bounded (PB) Components

• A component of the state vector is called
  polynomially bounded (PB) if the number of
  different values it can take is polynomially
  bounded in the size of the input.

The component for d is PB.
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Monotone Components
Monotone
Monotone
F11(27, 35, 2, 500, 1) ? 30, 35, 3, 530 ?
A non-PB component is called monotone if for each
stage j replacing its current value i by i gt i
cannot not decrease Fj, and PB components stay
unchanged in Fj(S) e.g., components 1, 2, and
4,
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Domination at Stage j
?
Monotone
?
PB

Monotone
?
for monotone
?
dom
if

for PB
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Theorem. B D is a pseudo-polynomial time
algorithm for a list scheduling problem if

1. There are a fixed number of components of the
   state vector
2. Each component is either monotone or PB
3. The objective is to minimize
4. Any constraint on a monotone component is a
   strict upper bound
5. All other constraints involve only PB components
6. Additional technical conditions (e.g., all
   functions can be computed in polynomial time.)

Proof. The number of undominated state vectors
at each stage is pseudo-polynomial.
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Moving from Pseudo-polynomial to an FPTAS.

• We want conditions under which Branch and
  ?-dominate leads to an FPTAS.
• We need to replace domination by ?-domination in
  all except one of the monotone components.

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Good and Bad Monotone Components

• A monotone component (say component 1) is good if
  all of the following conditions are satisfied
• It has no strict upper bound, and
• It cannot decrease from stage to stage.
• Non-Example Suppose we keep track of M1 M2
• Fj is not overly sensitive to small changes in s1
  e.g., Fj(s1(1 ?), s2, , sk, ) ? (1 n?)
  Fj(s1, s2, , sk).
• If it is not good, it is bad.

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On bad monotone components

• Condition 1. Any monotone component on which
  there are hard upper bounds is bad.
• Example. the processing time on machine 1 at
  stage j at most bj. So, M1 is bad.
• Small relative changes in the value of the
  component can mean the difference between the
  constraint being satisfied or not.

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On bad monotone components

• Condition 2
• Suppose s1 could decrease from stage to stage In
  this case, a small relative change in the value
  of s1 at some stage could have a very large
  impact on states at later stages. And so it is
  bad.
• e.g., Suppose we keep track of M1 M2

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On Bad Monotone States

• Condition 3. A state is bad if a small change in
  its value can cause a large change in the value
  at the next stage for some monotone state.

Suppose that we are minimizing total tardiness.
Bad Monotone
Bad Monotone
PB
Good Monotone
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?-Domination at Stage j
Bad Monotone
?
? (1 ?)
Good Monotone

PB
Good Monotone
? (1 ?)
? -domination is the same as domination except
for the (1 ?) term on the good monotone states.
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• Theorem. If a list scheduling problem satisfies
  conditions 1-6 from before, and if at most one
  monotone component is bad, then Branch and
  ?-dominate can be used to create an FPTAS.
• Note use ?-domination on all good monotone
  components, and use domination on the remaining
  monotone component.

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A list scheduling problem with outsourcing

• Schedule jobs on a single machine
• Each job j has a due date dj, and a maximum
  tardiness Lj
• job i can be outsourced at a cost of ci . At
  most K jobs can be outsourced.
• Objective minimize the weighted sum of the
  tardinesses of the jobs plus the outsourcing
  costs.
• subject to job j must be completed before dj
  Lj.

Processing Time on machine
bad-bad
States at stage j
Number outsourced
PB
Cumulative Objective Function
Good
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• Theorem. Branch and ?-dominate can be used to
  create an FPTAS for the list scheduling problem
  on the previous slide.

Note if we do not require a list schedule, the
previous problem is strongly NP-hard.
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References

• Contrast with Woeginger 2000
• very similar in essence
• fewer abstractions here more direct focus on the
  components of the transition function
• focus on list scheduling helps to clarify
  contribution
• different in generalizations that follow

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Other references

• Rich history of references in FPTASes
• Pioneers (1970s)
• Horowitz Sahni, Ibarra Kim, Sethi, Garey,
  Johnson, Babat, Gens, Levner, Lawler, Lenstra,
  Rinnooy Kan, and more
• 61 references in the ACM digital guide for FPTAS
• Domination in Branch and Bound is an old idea.
  But I dont have early references.

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Multi-criteria FPTAS
Suppose we have two or more objective criteria.

• We say that (c, b) ?-dominates (c, b) if
• c ? (1 ?) c and
• b ? (1 ?) b

A pareto set consists of a maximal number of
undominated solutions.
The size of a pareto set is polynomial in the
size of the data, 1/? and the number of
objectives.
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Multi-criteria FPTAS
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Multiple Criteria FPTAS

• Theorem. If a sequential problem is solvable
  with an FPTAS for one good objective using the
  results of Theorem 1, then the multiple criteria
  version is solvable with an FPTAS for multiple
  good objectives.
• Proof. Same argument as in single criterion case.

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References

• Multi-criteria FPTAS
• Hansen 1979
• Orlin 1981
• Warburton 1987
• Safer and Orlin 1995
• Papadimitriou and Yannakakis 2000, 2001
• Angel, Bampis, and Kononov 2003

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Machine Scheduling with crashing

• 2 machine scheduling problem
• Minimize the makespan
• Budget B
• Processing item j uses up some of the budget
• Processing time of item j is pj(b), where b are
  used.
• if b gt b, then pj(b) ? pj(b).
• Note there are an exponential number of possible
  decisions to make at stage j.
• We will modify BD.

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Machine Scheduling with crashing
Processing Time on machine 1
good
States at stage j
Processing Time on machine 2
good
Budget used up
bad
Place job j on machine 1 or 2
Decisions at stage j
How much budget to allocate
exp
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Branching at stage j. b budget allocated
Budget allocation as a binary decision process.
state
Exampleb ? 0, B/2
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Domination Rule
Suppose that the state Fj(S, 1, L)
?-dominates the state Fj(S, 1, U)
Then allocate a budget of L, and stop branching.
L
Stage j, state S, assign job j to machine 1,
budget assigned is between L and U
During the branching for budget allocation, stop
branching at a node denoted as L, U if
allocating the budget L gives a state that
?-dominates the state obtained by allocating U.
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Lemma and Theorem

• Lemma. In the binary expansion of the budget,
  the number of nodes starting from any state is
  polynomial in the size of the problem and in 1/?.
  
• Theorem. Branch and dominate is an FPTAS
  whenever it satisfies conditions 1 to 6 from
  before and whenever the number of bad monotone
  states is at most 1.

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Lot Sizing

• The previous analysis extends to dynamic
  lotsizing
• Branch and ?-dominate gives an FPTAS, even in the
  multiple criteria case.
• Extends work by
• Dada and Orlin 1981
• Safer and Orlin 1995
• Van Hoesel and Wagelmans 2001
• Safer, Orlin, and Dror 2003

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On FPTASs

• List Scheduling Problem
• polynomially bounded states
• monotone states, both good and bad
• constraints
• d-domination
• Extends and simplifies research by Woeginger