Facility Location using Facility Location using Linear Programming DualityLinear Programming Duality

Yinyu YeYinyu YeDepartment if Management Science and Department if Management Science and

EngineeringEngineeringStanford UniversityStanford University

InputInput• A set of clients A set of clients or citiesor cities D D

• A set of facilities A set of facilities F F withwith facility cost facility cost ffii

• Connection cost Connection cost CCijij, , (obey triangle (obey triangle inequality)inequality)

Output• A subset of facilities F’

• An assignment of clients to facilities in F’

Objective• Minimize the total cost (facility + connection)

Facility Location ProblemFacility Location Problem

Facility Location ProblemFacility Location Problem

location of a potential facility

client

(opening cost)

(connection cost)

Facility Location ProblemFacility Location Problem

location of a potential facility

client

(opening cost)

(connection cost)

cost connectioncost openingmin

R-Approximate Solution and Algorithm

:following thesatisfies that , cost, totalwith the

UFLP,ofsolution (integral) feasible a found algorithmAn

Cost

.1constant somefor

*

R

CostRCost

Hardness Hardness ResultsResults

NP-hard. Cornuejols, Nemhauser & Wolsey [1990].

1.463 polynomial approximation algorithm implies NP =P. Guha & Khuller [1998], Sviridenko [1998].

ILP Formulation

FiDjyx

FiDjyx

Djxts

yfxCMin

iij

iij

Fiij

Fi Dj Fiiiijij

,}1,0{,

,

1..

•Each client should be assigned to one facility.

•Clients can only be assigned to open facilities.

FiDjx

FiDjyx

Djxts

yfxCMin

ij

iij

Fiij

Fi Dj Fiiiijij

,0

,

1..

LP Relaxation and its Dual

FiDj

Fif

FiDjcts

Max

ij

iDj

ij

ijijj

Djj

,0

,..

Interpretation: clients share the cost to open a facility, and pay the connection cost.

.facility toclient ofon contributi theis },0max{ ijcijjij

Bi-Factor Dual Fitting

:following thesatisfies where,cost totalwith the

FLP, ofsolution (integral) feasible a found algorithman Suppose

jDj

j

FifR

FiDjcR

ifDj

ij

ijcijj

(2)

, )1(

.

: have then we0, and 1,constant somefor ** CRFRCF

RR

cfDj

j

ijfc

A bi-factor (Rf,Rc)-approximate algorithm is a max(Rf,Rc)-approximate algorithm

Simple Greedy Algorithm

Introduce a notion of time, such that each event can be associated with the time at which it happened. The algorithm start at time 0. Initially, all facilities are closed; all clients are unconnected; all set to 0. Let C=D

While , increase simultaneously for all , until one of the following events occurs:

(1). For some client , and a open facility , then connect client j to facility i and remove j from C;

(2). For some closed facility i, , then open

facility i, and connect client with to facility i, and remove j from C.

j

C j Cj

Cj ijj ci such that

Cj

iijj fc ),0max(

Cj ijj c

Jain et al [2003]

Time = 0Time = 0

F1=3 F2=4

3 5 4 3 6 4

Time = 1Time = 1

F1=3 F2=4

3 5 4 3 6 4

Time = 2Time = 2

F1=3 F2=4

3 5 4 3 6 4

Time = 3Time = 3

F1=3 F2=4

3 5 4 3 6 4

Time = 4Time = 4

F1=3 F2=4

3 5 4 3 6 4

Time = 5Time = 5

F1=3 F2=4

3 5 4 3 6 4

Time = 5Time = 5

F1=3 F2=4

3 5 4 3 6 4

Open the facility on left, and connect clients “green” and “red” to it.

Open the facility on left, and connect clients “green” and “red” to it.

Time = 6Time = 6

F1=3 F2=4

3 5 4 3 6 4

Continue increase the budget of client “blue”

Continue increase the budget of client “blue”

Time = 6Time = 6

The budget of “blue” now covers its connection cost to an opened facility; connect blue to it.

The budget of “blue” now covers its connection cost to an opened facility; connect blue to it.

F1=3 F2=4

3 5 4 3 6 4

5 5 6

The Bi-Factor Revealing LP

Given , is bounded above by

Subject to:

c

fR

k

jij

if

k

jj

1

1max

jl

iilj fc ),0max( ||21 D

ilijlj cc

cRfR

Jain et al [2003], Mahdian et al [2006]

alg. appr.-1.861 agot We.861.1 then ,861.1 cf RR

In particular, if

Approximation ResultsApproximation Results

Ratio Reference Algorithm1+ln(|D|) Hochbaum (1982) Greedy algorithm3.16 Shmoys et.al (1997) LP rounding2.408 Guha and Kuller (1998) LP rounding + Greedy augmentation1.736 Chudak (1998) LP rounding1.728 Charika and Guha (1999) LP + P-dual + Greedy augmentation1.61 Jain et.al (2003) Greedy algorithm1.517 Mahdian et.al (2006) Revised Greedy algorithm