recent advances in facility location optimization

8/6/2019 Recent Advances in Facility Location Optimization

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Recent advances in facility

location optimization

Binay K Bhattacharya

School of Computing ScienceSimon Fraser University


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What is facility locating?

GOOD

SERVICE

Clientneeds thiskind of service.

Facilityprovides somekind of service.


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Location problems in most general form can

be stated as follows:

a set ofclients originates demands for some kind

of goods or services.

the demands of the customers must be suppliedby one or more facilities.

the decision process must establish where to

locate the facilities.

issues like cost reduction, demand capture,equitable service supply, fast response time etc

drive the selection of facility placement.


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The basic elements of location models are:

a universe, U, from which a set Cof client inputpositions is selected,

a distance metric, d: UxU R+, defined over

the universe R+, an integer,p 1, denoting the number of

facilities to be located, and

an optimization function gthat takes as input a

set of client positions andp facility positions andreturns a function of their distances as measuredby the metric d.


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Problem statement

Select a set Fofp facility positions inuniverse U that minimizes g(F,C).


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Models of the universe of clients and

facilities

Continuous space

Universe is defined as a region, such that

clients and facilities may be placed

anywhere within the continuum, and the

number of possible locations is uncountablyinfinite.


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Discrete space

Universe is defined by a discrete set of

predefined positions.

Network space

Universe is defined by an undirected

weighted graph. Client positions are given

by the vertices. Facilities may be located

anywhere on the graph.


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Where should the radio tower be located?

Tower may be positioned anywhere

(continuous)

Tower may be positioned in five available slots

(discrete)

Tower may be located on the roadside

(network)


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Distance metric

Distance metric between two elements of the

universe further differentiates between specific

problems

Minkowski distance (Euclidean, Manhattan etc)

Network distance


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Optimization function : sum or maximum

The objective function is the most significant

characterization of a facility location problem.

p-centerproblem

Given a universe U, a set of points C, a

metric d, and a positive integerp, ap-centerof

C is a set ofp points Fof the universe Uthatminimizes

maxjC {miniFdij}.


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p-median problem

Given a universe U, a set of points C, a

metric d, and a positive integerp, ap-median

ofC is a set ofp points Fof the universe U thatminimizes

jC{miniFdij}.


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3-center

Euclidean p-center


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Euclidean p-center

(Megiddo and Supowit, 1984):p-center is NP-hard.

(Feder and Green 1988): -approximation remains NP-

hard for any < (1+7)/2 1.8229.

(Hwang, Lee and Chang 1993)p-center problem in R2

can be solved in O(np) time.


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Summary

Time complexities of algorithmic solutions to the Euclideanp-center problem

Drezner 1984, Hoffmann 2005 (arbitraryp and d = 1)

Megiddo 1983 (p=1, d=2)

Agarwal et al 1993, Chazelle et al. 1995 (p = 1 and d fixed)

Chan 1999 (p=2, d=2)

Agarwal and Sharir 1998 (p=2, darbitrary)

Agarwal and Procopuic 1998 (p fixed, darbitrary)


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1-median

Minimize the average distance to the facility

Euclideanp-median


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Euclideanp-median

1-median is not unique if the points are linearly dependent.

1-median is unique if the points are linearly independent

(Kupitz and Martini, 1997)

Difficult to compute 1-median in the plane.

Bajaj(1987, p 177) states there exist no exact algorithm under

models of computation where the root of an algebraic equation

is obtained using arithmetic operations and the extraction ofkth

roots

Chandrasekhar and Tamir (1990): a polynomial time

algorithm for an -approx of the 1-median.

Indyk (1999) and Bose et al (2003): linear in n and

polynomial in 1/ (1-median)


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Euclideanp-median

Hassin and Tamir(1991) O(pn) time solution for a line.

Suppowit and Meggido (1984): Problem is NP-hard in the

plane, and can not be approximated to within 3/2.

Summary of results


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Three problems

Mobile facility location problem.

p-median problem in tree networks

Approximation algorithms for network facility

location problems.


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MobileFacility

Location


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Collaborators

Sergey Bereg, Univ. of Texas at Dallas

Binay Bhattacharya, SFUStephane Durocher, UBCDavid Kirkpatrick, UBC

Michael Segal, Ben-Gurion Univ.


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(Static)Facility

Location

sites/clients


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(Static)Facility

Location

sites/clients


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StaticFacility

Location

sites/clientsfacilities/servers


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Centerproblems

1-center


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Centerproblems

1-center

Minimize the maximum distance to the facility


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Centerproblems

p-center

Minimize maximum distance to the closestfacility


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Medianproblems

1-median

Minimize the average distance to the facility


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DynamicFacility

Location


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DynamicFacility

Location


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DynamicFacility

Location


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DynamicFacility

Location


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discrete changes (insert/delete) focus on data structures to avoid

(full) recomputation


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MobileFacility

Location


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MobileFacility

Location


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MobileFacility

Location


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clients change position continuously

goal: update facility location(s) ina well-behaved fashion


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New constraints/criteria: facility locations should change

continuously (unlike exact 2-center)


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continuously (unlike exact 2-center) facilities may be subject to bounds

on velocity (unlike exact 1-center)


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p1

p2 p3


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p1

p2

p3p2

p3

c c


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p1

p2

p3p2

p3

c c

Possible to show that |cc|/|p2p2| > 2/|p2p2| v

provided |p2p2| 2/v2


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continuously (unlike exact 2-center) facilities may be subject to bounds

on velocity (unlike exact 1-center) quality of approximation


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continuously (unlike exact 2-centre) facilities may be subject to bounds

on velocity (unlike exact 1-centre) quality of approximation

simplicity of facility motion


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Mobile 1-center results

1-dimensional : Exact 1-center can bemaintained (need appropriate data structures)


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2-dimensional

Theorem: For any velocity v 0, there exist threesites such that a unitvelocity motion of thetwo of the sites induces an instantaneousvelocity > v of the Euclidean 1-center.


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2-dimensional (approximate)

Observation: Let 1, 2,.., nbe fixed nonnegativenumbers such that

1+ 2+..+ n =1.If all the

sitess1, s2, , sn move with velocity at most 1, 1s1+

2s2+..+ nsn also moves with velocity at most 1.


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Lemma: The center of mass of a set ofn sites provides

(2-2/n)-approximation of the Euclidean1

-center. Thefacility moves with velocity at most 1.


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Lemma: The center of mass of a set of n sites provides

(2-2/n)-approximation of the Euclidean 1-center. Thefacility moves with velocity at most 1.

Surprisingly, theaboveapproximation factorisasymptoticallyoptimal.


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Higher velocity approximation.

Bounding box center:


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Higher velocity approximation.

Bounding box center:Bounding box center moveswith velocity at most 2. Thecenter is (1+ 2)/2 approxi-mation of the Euclidean 1-

center.


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Summary: 2-approximation is realizable with velocity 1.

(1+2)/2-approximation is realizable with

velocity 2.


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velocity 2.

Question: Whatapproximation factorcanbe

achievedifthevelocityofthe facilityfisrestrictedtosome constantbetween1and2 ?


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velocity 2.

Question: Whatapproximation factorcanbe

achievedifthevelocityofthe facilityfisrestrictedtosome constantbetween1 and2 ?

A combination of the center of mass and the center of the

bounding box.


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(f1, v

1) : (location, velocity) of center of mass

(f2, v2) : (location, velocity) of bounding box center

Mixing strategy:(f, v) : (location, velocity) of the mixing center where

f = f1+ (1- ) f2, and

v = v1 + (1-)v2, 1 0.


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Upper Bound Lemma: For any > 0, there is a strategysuch that

(a) the approximation factor is (1+), and

(b) the velocity never exceeds (2+)(1+)/(2 + 2).


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Lower Bound Lemma: For any > 0, any (1+)-approximate mobile 1-center has velocity at least 1/(8).


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Approximations to the Euclidean 1-center

Strategy Approximations v max

Euclidean center 1 Center of mass 2 1

Bounding box (1+2)/2 1.2071 2 1.4142

Gaussian center 1.1153 4/ 1.2732


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Euclidean 1-median

The Euclidean 1-median moves with arbitrarily

high velocity.

The center of mass provides a (2-2/n)-

approximation using unit velocity There are examples where the approximation

ratio of the center of mass is arbitrarily close to 2.

An approximation ratio better than 2/3 to theEuclidean 1-median is impossible where the

facility is constrained to move no faster than the

clients.


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Open Problems

Provide tighter bounds for mobile 1-center

and 1-median problems.

k-center and k-median, k 2?

3-dimensions?

Clusterings?


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p-median problem in trees


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Vertex optimality of p-median

Theorem (Hakimi, 1965): There always exists an optimalp-median solution where the facilities are placedonly at the vertices of the network.


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Vertex optimality of p-median

Theorem (Hakimi, 1965): There always exists an optimalp-median solution where the facilities are placedonly at the vertices of the network.

x

u

v

via u

via v

f()

l([uv])

df(),u


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p


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p

g


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Open problems

The conjecture of Chrobak et al: gj(|Tx|) O(|Tx|).

If true, this will have significant impact on the complexity.

Can we remove the requirement thatp is fixed?

able to do when the tree is balanced

currently, not so when the tree is not unbalanced.

Extension of these to partial k-trees?

Summary ofproblems solved for tree


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Problem Vertex Current Prev.weights best best

p-median (const p) + O(nlogp+2n) O(pn2)3-median + O(nlog3n) O(n2)2-median (MWD) +/- O(nlogn) O(nlog2n)2-median (WMD) +/- O(nhlog2n) O(n3)

p-center (const p) + O(n) O(nlog2n)1-center +/- O(nlog3n) O(n2)

Collection depots

1-median + O(nlogn) O(n2)1-center + O(n) O(n2)

networks

C ll b t


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Collaborators

Boaz Ben Moshe

Robert Benkoczi

David Breton

Binay Bhattacharya

Qiaosheng Shi


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Discrete Mathematics (to appear)

MFCS 2003,

ESA 2005

ISAAC 2005

LATIN2006

unpublished

Papers


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Approximation algorithms

A -approximation algorithm for an optimization

problem is a polynomial-time algorithm that is

guaranteed to find a feasible solution of the

objective function value within a factor of of the

optimal.

The performance guarantee of the algorithm is .


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Three algorithmic techniques:

Primal-Dual: Use LP implicitly to find a solution

Local Search: Iteratively improving the integer solution

searching nearby solutions.

LP Rounding: Rounding fractional optimal solution

to nearbyoptimal integral solution.


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Overview of LP rounding

Formulate the problem as an IP problem.

Solve the corresponding LP-relaxation. LP-

relaxed solution is a lower bound on IP. Round the relaxed optimal solution.

Show that the rounding does not increase the

cost too much.


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Uncapacitated facility location problem

The universe is a network which is a complete

graph G.

The client set (C) and the facility set (F) are

subset of vertices ofG. Each facility iofFhas an opening cost fi if it is

open.

Problem is to select a subset of the facilities

such that the total cost to serve all the clients ofC is minimized.

Mathematical models


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Mathematical models

Notations used:

C: set of customers/clients.

F : set of candidate facility locations.wj: service demand of customerj.

fi : fixed cost of establishing a facility at location i.dij: per unit cost of servicing (distance of) customerjfrom

facility i.yi : decision variable which takes value 1 if facility iis

opened, otherwise it is 0.xij: customer js demand is supplied from facility i.

Integer programming formulation


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j

facilities clients

iXij=1



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j

facilities clients

iXij=1

Xij=0

i



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j

facilities clients

iXij=1

yi= 1


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Corresponding LP-relaxed formulation (Primal)


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Dual LP formulation

Interplay between the primal and dual


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p y p u

variables

Let (x*,y*) and (v*,*) be the optimal solutionof LP-Primal and LP-Dual respectively.

jCv*j= LP-Opt(strong duality theorem)

jCvj LP-Opt for any feasible v(weak duality

theorem)

Ifx*ij> 0implies cijev*j (closeness property)

A th t ( * *) d ( * *) k


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Assume that (x*,y*) and (v*,*) are known.

We can visualize the fractional solution of LP-Ras follows:

j

facilities clients

ix*ij> 0

ix*ij= 1 j

Sh T d d A d l (1997)


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Shmoys, Tardos and Aardal (1997)

j

facilities clients

ix*ij> 0

One iteration


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One iteration

j

facilities clients

ix*ij> 0

Select the remaining clientjwith minimum v*j.

Construct a cluster which

includes

One iteration


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One iteration

j

facilities clients

ix*ij> 0



includes

client j

One iteration


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One iteration

j

facilities clients

ix*ij> 0



includes

clientj

all facilities used

fractionally byj

One iteration


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O e te at o



includes

clientj

all facilities used

fractionally byj

all clients that use thisfacility (fractionally)

j

facilities clients

ix*ij> 0

One iteration


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action

j

facilities clients

i

Open the facility in the cluster

with the smallest opening cost.

Assign all the clients of the

cluster to the opened facility.

Observe that the unopened

facilities of this cluster are nevergoing to be in any other cluster.


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Assignment cost of a client k in the cluster

j

facilities clients

ix*ij> 0

k


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j

facilities clients

ix*ij> 0

k

i(j)


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j

facilities clients

ix*i(j)k > 0

k

k is directly connected

to facility (i(j))i(j)

Due to the closeness

property the assignment cost

of client k is at most v*k i.e.

ci(j)k v*k


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Assignment cost of a client kin the cluster

j

facilities clients

i

k

kis indirectly connected

to facility i(j)i(j)

i

From triangle inequality

ci(j)k ecik + cij+ ci(j)j

ev*k+ v*j+ v*je3v*k


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j

facilities clients

i

k

i(j)

i


ci(j)k ecik + cij+ ci(j)j

ev*k+ v*j+ v*j

e

3v*k

Total assignment cost is at

most 3jICv*j, which is at

most3*LP-opte3*IP-opt.

kis indirectly connected

to facility i(j)


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opening cost ofi(j): fi(j)

j

facilities clients

ix*ij> 0

i(j)

fi(j)= min fi

e fix*ij

e fiy*i

facilitiesused by j

facilities

used by j

facilities

used by j


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opening cost ofi(j): fi(j)

j

facilities clients

ix*ij> 0

i(j)

fi(j)= min fi

e fiy*i

open

facilities

facilities

used by j

Total opening cost is

fiy*i, which is at most

LP-opte IP-opt.

facilities

used by j


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Theorem: The LP-rounding algorithm is a 4-approximation algorithm for the uncapacitated

facility location problem.

Clustered randomized rounding


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(Chudak and Shmoys, 2003)

j

facilities clients

i

x*ij> 0

Choose the cluster centerin increasing v*j+ i Fcijx*ij,(letjbe the center)

In cluster centered atj,

open a facility iat randomwith probabiltyx*ij.

Open independently each

facility ithat is not contained

in the neighborhood of anycluster with probability y*i.

Assign each client to its

nearest open facility.

Clustered randomized rounding


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(Chudak and Shmoys, 2003)

Choose cluster center in

increasing v*j+ i F , sayj.

In cluster centered atj,open a facility iat random

with probabiltyx*ij.

Open independently each

facility ithat is contained in

the neighborhood of any

cluster with probability y*i.

Assign each client to its

nearest open facility.

Every client will have an

open facility in its neighbor-

hood (directly connected)

with large probability (> 1-1/e ).

Expected cost of the

solution is (1+2/e)*IP-opt.

Primal-Dual method


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Primal Dual method



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Corresponding LP-relaxed formulation

Dual LP formulation


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Complementary slackness (CS) property: Let (x*,y*) and(v*,*) be the optimal solution of LP-R and LP-Dualrespectively.

Primal CS: x*ij> 0implies v*j*ij= cij for all i, j.

y*i> 0implies j C*ij= fi for all i

Dual CS: v*j> 0implies i Fx*ij=1 forj

*ij> 0impliesx*ij= y*i for all i, j

Algorithm of Jain and Vazirani, 1999


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Algorithm of Jain and Vazirani, 1999

LP-R o r LP-Dual solution is not required.

First constructs a feasible dual solution (v,)

and then using the dual solution constructs aninteger feasible solution (x,y) of LP-R, andhence forIP.

(x,y) and (v,) satisfy the

Dual CS conditionand partially satisfy the Primal CS condition.

The algorithm is a 3-approximation algorithm.

Interpretation of the dual variables


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Suppose I Fand : C

Iis an optimal integralsolution. i.e. yi=1 iffi Iandxij= 1 iffi= (j).

Suppose (v,) denotes an optimal dual solution.

Ifi I, j Cij= fi .

Each open facility is fully paid for by the clients

using the facility.

Ifi= (j), vjij= cij.

We can think ofvjas the total price paid by clientj; of this cijgoes towards the use of edge (i,j) andij is the contribution ofjtowards the opening offacility i.

Some terminologies


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Some terminologies

A facility is fully paid when j Cij= fi.

A clientjhas reacheda facility iifvj cij.

If, in addition, iis fully paid,jgets connectedto i.

Algorithm to compute a feasible (v,)

( h 1)


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(phase 1)

Set vj 0;ij 0for all iandj.

UNTIL allj Care connected DO

-- increase vj for all unconnectedj.

-- increaseijfor all iandjsatisfying

jhas reached i

jis not yet connected

iis not fully paid.

c

b

a1

2

3

4

A facility is fully paid when j Cij= fi A clientjhas reacheda facility iifvj cij.

If, in addition, iis fully paid,jgets

connectedto i.


( h 1)


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(phase 1)



-- increase vjfor all unconnectedj.

--increaseijfor all iandjsatisfying

jhas reached i


iis not fully paid.


( h 1)


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(phase 1)



-- increase vjfor all unconnectedj.

-- increaseij

for all iandjsatisfying

jhas reached i


iis not fully paid. 1 a

A facility is fully paid when j Cij= fi A clientjhas reacheda facility iifvj

cij. If, in addition, iis fully paid,

jgets connectedto i.


( h 1)


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(phase 1)



-- increase vjfor all unconnected.


jhas reached i


iis not fully paid.

2

3

4

b

c

1a

A facility is fully paid when j Cij= fi A clientjhas reacheda facility iifvj

cij. If, in addition, iis fully paid,

jgets connected.


( h 1)


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c

(phase 1)





jhas reached i


iis not fully paid.

2

3

4

b1

a


(phase 1)


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(phase 1)





jhas reached i


iis not fully paid.

2

3

4

b

c

1a


(phase 1)


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(phase 1)





jhas reached i


iis not fully paid.

2

3

4

b

c

1 a

Algorithm to compute an integral (x,y)

(phase 2)


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(phase 2)

2

3

4

b

c

1 a

j

facilities clients

i ij > 0


(phase 2)


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(phase 2)

2

3

4

b

c

1 a

j

facilities clients

i ij > 0

i

j

iis fully paidbefore i.


(phase 2)


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(phase 2)

j

facilities clients

i ij > 0

i

j

Case:jis directly connected

to i, i.e.ij>0

From Phase 1 algorithm

vjij= cij

i.e.vi cij


(phase 2)


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(phase 2)

j

facilities clients

i ij > 0

i

j

Case:jis indirectly connected

to i.i.e.ij= 0.


cij cij+ cij+ cij.

Since the facility iis fully paid

before the facility i, vj

vj.

Therefore cii 3*vj.

One iteration


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Select the earliest fully paid

facility inot opened yet

(set yi=1).

Remove all clientsjwith ij >0(setxij=1)

j

facilities clients

iij> 0

One iteration


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Select the cheapest fully paid

facility inot opened yet

(set yi=1).

Remove all clientsjwith ij >0(setxij=1)

Remove all clientsjwherejis

indirectly connected to i.

(setxij

=1)

j

facilities clients

iij> 0

j

Relaxed complementary slackness (CS) property: Let (x,y)and (v ) be the solution determined during Phase 1 and 2


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and (v,) be the solution determined during Phase 1 and 2respectively.

Relaxed Primal CS: xij> 0implies 1/3cij vjij cij for all i, j.yi> 0implies j Cij= fifor all i

Dual CS: vj> 0implies i F xij=1 forj

ij> 0impliesxij= yi for all i, j

Theorem: The primal-dual algorithm of Jain and Vazirani is a

3-approximation algorithm.

Charikar and Guha(1999) showed that this algorithm alone

can not improved the solution any further.

Summary of UFLP approximation algorithms


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Generalizations


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p-median

- Charikar, Guha, Tardos and Shmoys

- Arya, Garg, Khandekar, Pandit,

Myerson and Munagala

- Jain and Vazirani

- Charikar and Guha

- Jain, Mahadian and Saberi

((1 + 2/e) lower bound)

Generalizations


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Capacitated facility location

Soft

-Shmoys, Tardos, Aardal

- Chudak and Shmoys

- Jain and Vazirani

- Korupolu, Rajaraman and Plaxton

- Chudak and Williamson

Generalizations


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Capacitated facility location

Hard

-Pal, Tardos and Wexler

Generalizations


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Fault tolerant facility location

-Jain and Vazirani

-Guha, Myerson and Munagala

- Swamy and Shmoys

- Jain, Mahdian, Markakis, Saberi,

Vazirani

Generalizations


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Facility location with penalties

- Charikar, Khullar, Mount and Narasimhan

- Jain, Mahdian, Marakis, Saberi and

Vazirani

Minimum sum of cluster diameters

..

Conclusions


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Rich source of problems requiring exact solution

requiring approximate solution

requiring fast solution

Developed tools have wider applications

scheduling

clustering

Most of the problems in a general network is NP-hard partial k-trees where k is small?


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Thank you

recent advances in facility location optimization

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