# LP Applications

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```					   Facility Location using
Linear Programming Duality
Yinyu Ye
Department if Management Science and
Engineering
Stanford University
Facility Location Problem
Input
• A set of clients or cities D
• A set of facilities F with facility cost fi
• Connection cost Cij, (obey triangle inequality)
Output
• A subset of facilities F’
• An assignment of clients to facilities in F’
Objective
• Minimize the total cost (facility + connection)
Facility Location Problem



location of a
   potential facility

                 (opening cost)

          client
           (connection cost)
Facility Location Problem



location of a
        potential facility

                      (opening cost)

               client
                (connection cost)

min    opening cost  connection cost
R-Approximate Solution
and Algorithm
An algorithm found a feasible (integral) solution of UFLP,
with the total cost, Cost , that satisfies the following :

Cost  R  Cost *
for some constant R  1.
Hardness Results

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

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

Min         C
iF jD
ij   xij   f i yi
iF

s.t.       x
iF
ij   1           jD

xij  yi                j  D,  i  F
xij , yi {0,1}  j  D,  i  F

•Each client should be assigned to one facility.

•Clients can only be assigned to open facilities.
LP Relaxation and its Dual

Min       C x   f y
iF jD
ij ij
iF
i i           Max     
jD
j

s.t.    x    ij   1        jD               s.t.    j   ij  cij  j  D,  i  F
  f
iF
iF
xij  yi             j  D,  i  F           jD
ij   i

xij  0              j  D,  i  F            ij  0        j  D,  i  F
Interpretation: clients share the
cost to open a facility, and pay
the connection cost.
ij max{0,  j  cij } is the contributi on of client j to facility i.
Bi-Factor Dual Fitting
Suppose an algorithm found a feasible (integral) solution of FLP,
with the total cost   j , where  j satisfies the following :
jD

(1)            j   ij  Rc cij          j  D,  i  F
(2)                
jD
ij    R f fi        iF

for some constant Rc , R f  1 and  ij  0, then we have :
F  C    j  R f F *  Rc C * .
jD

A bi-factor (Rf,Rc)-approximate algorithm is a
max(Rf,Rc)-approximate algorithm
Simple Greedy Algorithm
Jain et al [2003]
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  j set to 0. Let C=D

While C   , increase  j simultaneously for all j  C , until
one of the following events occurs:

(1). For some client j  C , and a open facility i such that  j  cij
, then connect client j to facility i and remove j from C;

(2). For some closed facility i,  max( 0, j  cij )  f i , then open
jC

facility i, and connect client j  C with  j  cij to facility i, and
remove j from C.
Time = 0

F1=3                        F2=4

3   5   4    3   6          4
Time = 1

F1=3                        F2=4

3   5   4    3   6          4
Time = 2

F1=3                        F2=4

3   5   4    3   6          4
Time = 3

F1=3                        F2=4

3   5   4    3   6          4
Time = 4

F1=3                        F2=4

3   5   4    3   6          4
Time = 5

F1=3                        F2=4

3   5   4    3   6          4
Time = 5

F1=3                        F2=4

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

3   5   4    3   6           4
Time = 6

F1=3                        F2=4

Continue
increase the
budget of client
“blue”

3   5   4    3   6           4
Time = 6

F1=3                               F2=4

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

3       5   4       3   6            4

5           5                        6
The Bi-Factor Revealing LP
Jain et al [2003], Mahdian et al [2006]

Given R f , Rc is bounded above by
k


j 1
j    R f fi
max                       k

c
j 1
ij

Subject to:     1  2    | D|
 max( 0,
l j
j    cil )  f i

 j  l  cij  cil

In particular, if
R f  1.861, then Rc  1.861. We got a 1.861 - appr. alg.
Approximation Results

Ratio        Reference                     Algorithm
1+ln(|D|)  Hochbaum (1982)             Greedy algorithm
3.16    Shmoys et.al (1997)             LP rounding
2.408 Guha and Kuller (1998) LP rounding + Greedy augmentation
1.736      Chudak (1998)                 LP rounding
1.728 Charika and Guha (1999) LP + P-dual + Greedy augmentation
1.61      Jain et.al (2003)          Greedy algorithm
1.517    Mahdian et.al (2006)     Revised Greedy algorithm

```
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