The Online Transportation Problem

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The Online Transportation Problem

• Christine Chung
• Kirk Pruhs
• Patchrawat Uthaisombut

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The Online Transportation Problem
5 miles
3 miles
4 miles
2 miles
2 miles
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The Online Transportation Problem

• Given
• a set of hospitals, each with a different sized
  ambulance fleet
• emergency sites that arise in unpredictable
  locations over time, each in immediate need of an
  ambulance
• Return
• an assignment of ambulances to emergencies that
  minimizes the total distance traveled by all the
  ambulances
• To measure algorithm goodness
• ON(I) is the value of our online algorithms
  solution and OPT(I) is the value of the optimal
  offline solution
• competitive ratio

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Why Greedy can be Bad

• Consider A, a large hospital with a dozen
  available ambulances, and B, a small hospital
  with only 2 ambulances.
• Imagine an emergency breaks out between A and B,
  but a bit closer to B.

A
B
9 miles
10 miles
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Why Greedy can be Bad

• Since B is closer, we dispatch an ambulance from
  B.
• But what if two emergencies then break out on the
  other side of B from A?
• This solution costs 31 miles!

A
A
B
20 miles
9 miles
2 miles
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Why Greedy can be Bad

• If instead we chose hospital A for the first
  emergency,
• then used the ambulances at hospital B for the
  second and third...
• ...the distance traveled is much less!

A
A
B
2 miles
10 miles
2 miles
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Challenges Faced

• Intuition A good algorithm must find a balance
  between distance of an emergency from the
  hospitals and the resources remaining at each
  hospital.
• Negative Results Such balancing is not possible.
  The competitive ratio of every deterministic
  online algorithm is bad (linear in the number of
  hospitals).

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Our Contributions

• Use Linear Programming Duality and Dual Fitting
  as tools to balance the distance traveled and
  ambulance reserve at each hospital.
• Use Resource Augmentation
• Compare our online solution to the optimal
  solution with one less ambulance per hospital.
• Intuitively one less ambulance per hospital
  should not usually change the optimal solution
  very much, so

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A Pair of Linear Programs

• minimize
• subject to

P
P
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The Duals

• maximize
• subject to

D
D
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Dual Fitting


PRIMAL FEASIBLE
DUAL FEASIBLE
OPT
P
DP/F
cost of solution

• As emergencies arise, dual fitting maintains a
  primal solution P and a dual solution D such
  that DP/F.
• This is a certificate that P is an F-
  approximation of the offline OPT.

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Dual Fitting with Resource Augmentation

PRIMAL FEASIBLE
DUAL FEASIBLE


No extra ambulances
OPT
P
DP/F
cost of solution
DP/F


cost of solution
Extra ambulances
OPT
P
D

• We maintain a primal P and dual D solutions such
  that DP/F
• This is a certificate that P is an F F
  approximation to the optimal solution without
  extra ambulances.

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Algorithm Description

• For each emergency j that arrives
• Let i be the hospital with available ambulances
  that minimizes dij ?i
• Assign emergency j to hospital i
• Set the dual variable ?j dij ?i
• We increase the dual variables ?i for those
  hospitals that had dij ?i dij ?i
• Increase ?i enough to maintain DP/F
  relationship.
• Intuition
• The dual variables ?i grow as a hospital depletes
  its ambulances, and grow more quickly if a
  hospital has a small fleet.
• A high value of ?i makes it less likely that
  hospital i will be selected.

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Current Work

• We can obtain poly-logarithmic competitiveness
  using resource augmentation and metric embedding
  of hierarchically well-separated trees, but this
  algorithm is not as elegant or practical as our
  dual fitting algorithm.
• We are trying to analyze the dual fitting
  algorithm.