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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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