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

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Collection Depots Facility Location Problems in Trees R. Benkoczi, B. Bhattacharya, A. Tamir. 陳冠伶 ‧ 王湘叡 ‧ 李佳霖 ‧ 張經略 Jun 12, 2007. Outline. By 陳冠伶. INTRODUCTION. C lient (demand service). Settings. F acility (service center). C ollection D epots. Cost of Service Trip. F. P 2. D.

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

Jun 12, 2007

Client (demand service)

Settings

Facility (service center)

Collection Depots

Cost of Service Trip

F

P2

D

P1

Service Cost

2(P1+P2)‧w(c)

C

Application (1)

Express Transportation

Application (2)

Garbage collection

Problem
• IN: given a tree and
• points of clients
• points of collection depots
• an integer k
• OUT
• Optimal placements of kfacilities
• that minimizes some global function of the service cost for all clients.
Objective – Minimax
• Minimize the service cost of the most expensive client

C

C

D

D

D

C

D

F

D

C

C

Minimax – center problems

1-center

Minimize the maximum distance to the facility

Minimax – center problems

k-center

Minimize the maximum distance to the closest facility

Objective – Minisum
• Minimize the total service cost

C

C

D

D

D

C

D

F

D

C

C

Minisum – median problems

1-median

Minimize the average distance to the facility

Minisum – median problems

k-median

Minimize the average distance to the closest facility

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
Prune and Search
• Every iteration, eliminate a fraction of impossible instances.
• Binary Search
• T(n)=T(n/2)+1
• T(n)=O(lg n)
Observation
• c(f)=max min r(f, vi)
• Service cost is non-decreasing when the facility goes away from the client.
Where could the facility be?
• A linear time algorithm could determine!

T1

T2

Ti

Tk

Initial tree

depot

client

Divide T(i) into S1 and S2
• Find the centroid and partition the tree into two parts

centroid

• S1 > 1/3 |T(i)|
• S2 > 1/3 |T(i)|
Find the Xmax
• Find the client Xmax with the largest service cost from the centroid.

Xmax

f

S2

S1

foptmust be in S1

Xmax

Special case
• Centroid is the optimal

Should be optimal

X’max

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

δmed

• S2
Partition S1 by δmed
• Find all f’, they form trees T1, T2, …,Tn
• There are two cases, fopt is in ∪Tior not

f

T1

T2

T3

fopt is in ∪Ti
• If fopt in red, consider K+, δ(fopt)<δmed<δ(K+)
• For a facility F’ in S1 and a client in S2, δ(fopt, u) is in S1

δ(f’, u)

δ(f’, u)

fopt

f’

fopt is in not ∪Ti
• If fopt is not in red, consider K-,

δ(K-)<δmed <δ(fopt)

• For a facility F’ in Sand a client in S2, δ(fopt, u) is in S2
• Similar to previous

case

• Only fopt in ∪Ti is considered.

δ(fopt, u)

f’

fopt

Details on fopt is in ∪Ti
• Arbitrarily paired clients in K+
• For each pair (u, v), Compute tuv s.t. w(v)(tuv+d(c,v))=w(u).(tuv+d(c,u))
• Compare tmed and

d(fopt, c)+d((fopt,c),p(fopt, c))

δ(f’, u)

fopt

fopt

d(fopt, c)+d((fopt,c),p(fopt, c)) < tmed
• consider tmed<tuv
• d(fopt, c)+d((fopt,c),p(fopt, c))<tmed<tuv

δ(f’, u)

fopt

fopt

d(fopt, c)+d((fopt,c),p(fopt, c)) > tmed
• consider tmed>tuv
• d(fopt, c)+d((fopt,c),p(fopt, c))>tmed>tuv
• ¼ K+ can be removed

δ(f’, u)

fopt

fopt

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: Ο(n2)
• Using Spine decomposition and pre-sorting: Ο(nlogn)
Cost of Subtree

d

d2

v

dnew

d4

f

c2

d1

cj

d3

c1

c3

c4

Complexity
• Construction for the SD has time complexity Ο(n) and space complexity Ο(n)
• Costs of the subtrees can be evaluated in constant timeonce j is determined.
• If we use binary search with dnew, we spend Ο(logn) time for every subtree. So Ο(log2n).
• Use the sequential search in sorted order. So Ο(logn).
• The 1-median collection depots problem in tree can be sloved in Ο(nlogn) time and Ο(n) space.
The objective
• To minimize the sum of facility opening costs plus service costs for servicing the clients.
The “自給自足” property (1/4)
• We fixed an arbitrary optimal solution and explore its structure.
The “自給自足” property (2/4)
• Consider an arbitrary vertex v.
• xv: minimize the trip cost of serving v
• yv:be a closest facility to v.

yv

Assumed (for contradiction) servicing facility for client C

xv

v

Tleft

Tright

client C

The “自給自足” property (3/4)

Assumed (for contradiction) servicing facility for client C

yv

xv

v

client C

Tright

Tleft

The “自給自足” property (4/4)
• The blue part of the following graph is proven by symmetry.

yv

xv

v

Tleft

Tright

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

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!
The technical things
• Due to some complications, the final Dynamic Programming is much more complicated…
• But the proof requires no special technique beyond the “自給自足” property.
• The challenge is to devise the “right” recurrences to carry out the aforementioned intuitive approach.
Time complexity
• Easily verified to be polynomial.