Collection Depots Facility Location Problems in Trees R. Benkoczi, B. Bhattacharya, A. Tamir

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Collection Depots Facility LocationProblems in
TreesR. Benkoczi, B. Bhattacharya, A. Tamir

•   ???????????????
• Jun 12, 2007

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Outline
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INTRODUCTION

• By ???

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Settings
Client (demand service)
Facility (service center)
Collection Depots
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Cost of Service Trip
F
P2
D
P1
Service Cost
2(P1P2)?w(c)
C
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Application (1)
Express Transportation
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Application (2)
Garbage collection
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Problem

• IN given a tree and
• points of clients
• points of collection depots
• an integer k
• OUT
• Optimal placements of k facilities
• that minimizes some global function of the
  service cost for all clients.

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Objective Minimax

• Minimize the service cost of the most expensive
  client

C
C
D
D
D
C
F
D
D
C
C
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Minimax center problems
1-center
Minimize the maximum distance to the facility
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Minimax center problems
k-center
Minimize the maximum distance to the closest
facility
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Objective Minisum

• Minimize the total service cost

C
C
D
D
D
C
F
D
D
C
C
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Minisum median problems
1-median
Minimize the average distance to the facility
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Minisum median problems
k-median
Minimize the average distance to the closest
facility
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Classifications
Classification Condition Type
Location of facility tree node Discrete
Location of facility tree edge Continuous
Weight of client included Weighted
Weight of client excluded Unweighted
Availability of depot some Restricted
Availability of depot all Unrestricted
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Summary of Results

• Unrestricted 1-center problem
• O(n)
• Unrestricted median problems
• 1-median O(nlogn)
• k-median O(kn3)
• Restricted k-median problem
• NP-complete
• Facility setup costs are not identical

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1-CENTER PROBLEM

• BY ???

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Prune and Search

• Every iteration, eliminate a fraction of
  impossible instances.
• Binary Search
• T(n)T(n/2)1
• T(n)O(lg n)
• How about

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Observation

• c(f)max min r(f, vi)
• Service cost is non-decreasing when the facility
  goes away from the client.

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Where could the facility be?

• A linear time algorithm could determine!

T1
T2
Ti
Tk
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Initial tree
depot
client
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Divide T(i) into S1 and S2

• Find the centroid and partition the tree into two
  parts

centroid
S1 gt 1/3 T(i)
S2 gt 1/3 T(i)
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Find the Xmax

• Find the client Xmax with the largest service
  cost from the centroid.

Xmax
f
S2
S1
fopt must be in S1
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Special case

• Centroid is the optimal

Should be optimal
Xmax
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Partition the clients

• Compute all depot distance
• Find the median dmed
• Separate all clients into two sets, K (red) and
  K- (blue)

dmed
S2
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• Consider f in S1, that depot distance d(f)lt
  dmed

d(f)lt dmed
f
S1
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Partition S1 by dmed

• Find all f, they form trees T1, T2, ,Tn
• There are two cases, fopt is in ?Ti or not

f
T1
T2
T3
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fopt is in ?Ti

• If fopt in red, consider K, d(fopt)ltdmedltd(K)
• For a facility F in S1 and a client in S2,
  d(fopt, u) is in S1

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fopt is in not ?Ti

• If fopt is not in red, consider K-,
• d(K-)ltdmed ltd(fopt)
• For a facility F in S and a client in S2,
  d(fopt, u) is in S2
• Similar to previous
• case
• Only fopt in ?Ti is considered.

d(fopt, u)
f
fopt
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Details on fopt is in ?Ti

• Arbitrarily paired clients in K
• For each pair (u, v), Compute tuv s.t.
  w(v)(tuvd(c,v))w(u).(tuvd(c,u))
• Compare tmed and
• d(fopt, c)d((fopt,c),p(fopt, c))

d(f, u)
fopt
fopt
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d(fopt, c)d((fopt,c),p(fopt, c)) lt tmed

• consider tmedlttuv
• d(fopt, c)d((fopt,c),p(fopt, c))lttmedlttuv

d(f, u)
fopt
fopt
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d(fopt, c)d((fopt,c),p(fopt, c)) gt tmed

• consider tmedgttuv
• d(fopt, c)d((fopt,c),p(fopt, c))gttmedgttuv
• ¼ K can be removed

d(f, u)
fopt
fopt
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1-MEDIAN PROBLEM

• BY ???

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The 1-median Problem

• Find a placement for facility to minimize the
  cost of all tours.
• i.e. minimize the sum of weighted distances of
  the facility to client, then to optimal depot,
  and return to facility.
• For the path of a facility to a client, the
  closest depot can be found efficiently.
• Brute Force ?(n2)
• Using Spine decomposition and pre-sorting
  ?(nlogn)

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The Spine Decomposition
r0
5
3
3
3
2
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Construct Search Tree
r0
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Search Tree of SD
r0
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Super-path of Search Tree
r0
f
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Cost of Subtree
d
d2
v
dnew
d4
f
c2
cj
d1
d3
c1
c3
c4
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Complexity

• Construction for the SD has time complexity ?(n)
  and space complexity ?(n)
• Costs of the subtrees can be evaluated in
  constant time once j is determined.
• If we use binary search with dnew, we spend
  ?(logn) time for every subtree. So ?(log2n).
• Use the sequential search in sorted order. So
  ?(logn).
• The 1-median collection depots problem in tree
  can be sloved in ?(nlogn) time and ?(n) space.

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UNRESTRICTEDK-MEDIAN PROBLEM

• BY ???

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The objective

• To minimize the sum of facility opening costs
  plus service costs for servicing the clients.

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The ???? property (1/4)

• We fixed an arbitrary optimal solution and
  explore its structure.

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The ???? property (2/4)

• Consider an arbitrary vertex v.
• xv minimize the trip cost of serving v
• yvbe a closest facility to v.

yv
Assumed (for contradiction) servicing facility
for client C
xv
v
Tleft
Tright
client C
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The ???? property (3/4)
Assumed (for contradiction) servicing facility
for client C
yv
xv
v
client C
Tright
Tleft
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The ???? property (4/4)

• The blue part of the following graph is proven by
  symmetry.

yv
xv
v
Tleft
Tright
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The intuition (1/2)

• The total cost can be partitioned into four
  categories the red, yellow, blue cost and v.

yv
xv
v
Tleft
Tright
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The intuition (2/2)

• The optimal solution has to be a combination of
  optimal substructures
• You have to be optimal in the red (to minimize
  the red cost) and the yellow (to minimize the
  yellow cost).
• This almost leads to Dynamic Programming already!

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The technical things

• Due to some complications, the final Dynamic
  Programming is much more complicated
• But the proof requires no special technique
  beyond the ???? property.
• The challenge is to devise the right
  recurrences to carry out the aforementioned
  intuitive approach.

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Simple intuition, complicated recurrences take a
look
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Time complexity

• Easily verified to be polynomial.