APPROXIMATION ALGORITHMS FOR
FACILITY LOCATION PROBLEMS
David B. Shmoys, Eva Tardos, Karen Aardal
Twenty-Ninth annual ACM symposium on Theory of computing (STOC),
1997
Presentation Prepared by Sapna Grover
ALGORITHM
Assignment of
clients to
facilities
4 ^
3C j
1/6
2/3
1/6
2/3
1/12
1/3
1/12
1/3
Opening variables, yi
Assignment
variables, xij
(x1,y1) is the LP Solution
^
C
is the Connection cost
that client j is paying in
(x1,y1)
j
1/4
^
C
R
1/5
4/5
1/3
4/3
j
  d j xij cij
iF
1/2
Opening facilities to 4 Now,
timesevery
the client is being served by facilities
current extent and thereby
within
its ball.
Note that
at least 1/4th extent of dj is being
changing the assignments
New
Solution
= (x2,y2) within its ball
served
from facilities
correspondingly
Cf (x2,y2) ≤ 4 Cf (x1,y1)
Cs (x2,y2) ≤ Cs (x1,y1)
• Cf (x,y) = facility opening cost of solution
(x,y)
• Cs (x,y) = service cost of solution (x,y)
• C(x,y) = ^total cost of solution (x,y)
• R=4/3 C
j
• At least 1/4th extent of dj is being served
from facilities within its ball
• Note: Fields in red designate opening
variables of facilities and fields in purple
indicate assignment of clients to facilities.
^
And within
the with
facility
i* withC
smallest
opening cost
Consider
theitball
smallest
j
Open i* and close all others.
0
1
0
r
j
New Solution = (x3,y3)
• Bounding
Cf (x3,y3) ≤Service
Cf (x2,y2Cost
) ≤ 4in
Cf (x
(x31,y
,y31),)
i*
r
≤3,y
cj’i’3)+≤c4C
cji*1,y(By
• cCj’i*
i’j +
s (x
s (x
1) tirangle ineq.)
≤ RC(x
+ 2r
• Thus,
3,y3) ≤ 4 C(x1,y1) ≤ 4 IPOPT ,
0
≤ 3R ^ solution.
a 4-factor
i’
≤ 4 Cj
1/4+1/4
R
Thus, Cs (x3,y3) ≤ 4Cs (x1,y1)
• (x1,y1) is the LP Solution
+1/2 =1
1/4
Assign j’ completely to i*.
• Cf (x,y) = facility opening cost of
solution (x,y)
R
j’
R
• Cs (x,y) = service cost of solution (x,y)
This will dissolve ball j’.
1/4
• C(x,y) = ^total cost of solution (x,y)
1/2
• R=4/3 C j
• At least 1/4th extent of dj is being
served from facilities within its ball.
Now consider another client j’, being served by a facility i’, recently
• Cf (x2,y2) ≤ 4 Cf (x1,y1)
closed in ball of j.
• Cs (x2,y2) ≤ Cs (x1,y1)
Note that i’ lies in overlapping region of j and j’.
Questions?
ALGORITHM
(x1,y1) is the LP Solution
^
C
4 ^
3C j
is the Connection cost
that client j is paying in
(x1,y1)
j
^
C
R
j
  d j xij cij
iF
• Cf (x,y) = facility opening cost of solution
(x,y)
• Cs (x,y) = service cost of solution (x,y)
• C(x,y) = ^total cost of solution (x,y)
• R=4/3 C
j
• At least 1/4th extent of dj is being served
from facilities within its ball
At least 1/4th extent of dj is being served
from facilities within its ball