PRIMAL-DUAL APPROXIMATION ALGORITHMS FOR METRIC FACILITY LO - PowerPoint Presentation

Slide1

PRIMAL-DUAL APPROXIMATION ALGORITHMS FOR METRIC FACILITY LOCATION

AND K-MEDIAN PROBLEMS

K. Jain V.

Vazirani

Journal of

the

ACM, 2001

 Slide2

PRIMAL-DUAL APPROACH

We start by constructing a primal problem.

On the basis of the primal problem, a dual problem is constructed.

We solve the dual problem while maintaining its feasibility in order to find a feasible primal solution.Slide3

INTEGER PROGRAMMING PROBLEM

min

Σ

j

ε

C

Σ

i

ε

F

c

ij

x

j

+

Σ

i

ε

F

f

i

y

i

s.t

Σ

i

ε

F

x

ij

≥ 1 for all j

ε

C

y

i

-

x

ij

≥ 0 for all

i

ε

F, j

ε

C

x

ij

ε

{0,1} for all

i

ε

F, j

ε

C

y

i

ε

{0,1} for all

i

ε

F

y

i

denotes whether facility

i

is open or not

x

ij

denotes whether client j is connected to a facility

i

or not

The first constraint ensures that each city is connected to at least one facility.

The second constraint ensures that if a client is assigned to a facility then that facility must be open. Slide4

RELAXED LP PRIMAL PROBLEM

min

Σ

j

ε

C

Σ

i

ε

F

c

ij

x

j

+

Σ

i

ε

F

f

i

y

i

s.t

Σ

i

ε

F

x

ij

≥ 1 for all j

ε

C

y

i

-

x

ij

≥ 0 for all i

ε

F, j

ε

C

0 ≤

x

ij

≤ 1

for all i

ε

F, j

ε

C

0

≤

y

i

≤ 1

for all i

ε

FSlide5

DUAL PROBLEM

max

Σ

j

ε

C

α

j

s.t

α

j

-

β

ij

≤

c

ij

for all

i

ε

F, j

ε

C

Σ

j

ε

C

β

ij

≤ f

i

for all

i

ε

F

α

j

≥ 0 for all j

ε

C

β

ij

≥ 0 for all

i

ε

F, j

ε

C

α

j

is the total price paid by a client j.

α

j

=

c

ij

+

β

ij

A part of

α

j

goes towards paying the service cost of facility

i

, i.e.,

c

ij

and

β

ij

is the contribution of j towards the facility opening cost of

i

.Slide6

COMPLIMENTARY SLACKNESS CONDITIONS

For Primal

for all i

ε

F, j

ε

C

:

x

ij

> 0 ═> αj - βij = cijfor all i ε F: yi > 0 ═> Σj ε C βij = fi

For dual

for all j

ε

C:

α

j

> 0 ═>

Σ

i

ε

F

x

ij

= 1

for all i

ε

F, j

ε

C:

β

ij

> 0 ═>

y

i

=

x

ij

Slide7

TERMS USED IN THE PAPER

Tight edge

:- An edge (

i,j

) between a client j and a facility

i

becomes tight when

α

j

=

cij. Special edge:- An edge (i,j) between a client j and a facility i is special if βij > 0.Unconnected Client:- A client j is said to be unconnected with a facility i if edge between them is not tight. Connected Client:- A client j is said to be connected with a facility i if edge between them is tight. Connecting Witness

:- A facility

i

is known as connecting witness for all those clients with whom it shares a tight edge.Slide8

ALGORITHM

Phase I

:-

Initially all the clients are unconnected and

α

j

= 0 for all the clients.

Raise the dual variables

α

j for each unconnected client j uniformly at a unit rate until an edge between some facility i and client j becomes tight.As soon as an edge (i,j) between client j and a facility i becomes tight, start raising βij at the same rate as αjA facility is declared temporarily open when its opening cost has been completely paid, i.e., Σj βij = fi.Slide9

As soon as a facility is declared temporarily open, all the clients that share tight edges with this facility are declared connected and the facility becomes its connecting witness.

All such edges for which

β

ij

> 0 are declared special.

If a client j gets a tight edge to an already open facility i, j is declared as connected to i, i becomes its connecting witness and its

β

ij

is not raised.

Phase I continues until all the clients become connected.Slide10

At t = 0

α

1

=0

α

3

=0

2 9 8 6 2 7

f

1=1 f2=1 f3=6 4 4 5 α2=0Slide11

At t = 1

α

1

=1

α

3

=1

2 9 8

6

2 7 f1=1 f2=1 f3=6 4 4 5 α2=1Slide12

At t = 2

α

1

=2

α

3

=2

2

9

8 6 2 7 β11=0 β23=0 f1=1 f2=1 f3=6 4 4 5 α2=2

Tight edge: Slide13

At t = 3

α

1

=3

α

3

=3

2 9 8 6 2 7

β11=1 β23=1 f1=1 f2=1 f3=6 4 4 5 α2=3

Tight edge:

Special edge:Slide14

At t = 4

α

1

=3

α

3

=3

2 9 8 6 2 7

β11=1 β23=1 f1=1 f2=1 f3=6 4 4 5 α2=4

Tight edge:

Special edge: Slide15

At the end of Phase 1

2 2

f

1

=1 f

2

=1

4 4 Tight edge:Special edge: Slide16

Phase II:-

It consist of two parts.

Part 1:- To choose facilities that will opened permanently.

Part 2:- To connect every client to exactly one facility.

Slide17

PART 1

:-

Let F

t

denote the set of all temporarily open facilities.

Consider a graph G with all the special edges of path length at most 2, induced on F

t

.

Construct a maximal independent set I of G.

Declare all the facilities in I as permanently open.Slide18

Phase 2 - part 1:- Forming independent sets

2 2

f

1

=1

f

2=1 4 4

Permanent FacilitySlide19

PART 2

:-

Case 1:- Assign client j to the facility

i

є

I with which it shares a special edge. Declare client j as directly connected to it.

Case 2:- If client j does not have a special edge to any facility

i

є I then assign j to a facility with which it shares a tight edge such that i was its connecting witness. Declare client j as directly connected to i.Case 3:- If client j has neither a special edge nor a tight edge with any facility in I , then assign j to a facility i* which is a neighbor of i in the graph G such that i* є I. Declare client j as indirectly connected to i*.Slide20

Phase 2 - part 2- Assigning

clients

to facilities

Case 1: Directly

Connected

Case 2: Directly

Connected Case 3: IndirectlyConnected

=

Directly

connected

=

Indirectly

connected Slide21

ANALYSIS

Let

α

j

=

α

j

f

+

α

jc, where αjf = βij and αjc = cijΣi ε I fi

=

Σ

j

ε

C

α

j

f

…

eqn

I

Consider an indirectly connected client j to

i

c

ij

≤ 3*

α

j

c

…

eqn

II

Adding the results of

eqn

I

(multiplied

by

3)

and

eqn

II

,

we get,

Σ

i

ε

I

Σ

j

ε

C

c

ij

x

ij

+ 3*

Σ

i

ε

I

f

i

y

i

≤ 3*

Σ

j

ε

C

α

j