Facility Location using Linear Programming Duality Yinyu Ye Department if Management Science and...

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