1. 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

2. ALGORITHM Assignment of clients to facilities 1/12
Opening variables, yi
(x1,y1) is the LP Solution
1/3
Assignment variables, xij
is the Connection cost that client j is paying in (x1,y1)
1/12
1/6
1/6
1/3
2/3
2/3
1/4 R
1/5
1/3
4/5
1/2
4/3
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
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.
Opening facilities to 4 times the current extent and thereby changing the assignments correspondingly
Now, every client is being served by facilities within its ball.
New Solution = (x2,y2)
Cf (x2,y2) ≤ 4 Cf (x1,y1)
Cs (x2,y2) ≤ Cs (x1,y1)
Note that at least 1/4th extent of dj is being served from facilities within its ball

3. Consider the ball with smallest
And within it the facility i* with smallest opening cost
Open i* and close all others. 1 r j
New Solution = (x3,y3)
Bounding Service Cost in (x3,y3),
cj’i* ≤ cj’i’ + ci’j + cji* (By tirangle ineq.)
≤ R + 2r
≤ 3R
≤ 4
Thus, Cs (x3,y3) ≤ 4Cs (x1,y1) i*
Cf (x3,y3) ≤ Cf (x2,y2) ≤ 4 Cf (x1,y1)
Cs (x3,y3) ≤ 4Cs (x1,y1)
Thus, C(x3,y3) ≤ 4 C(x1,y1) ≤ 4 IPOPT , a 4-factor solution. r i’
1/4+1/4
+1/2 =1 R
(x1,y1) is the LP Solution
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
At least 1/4th extent of dj is being served from facilities within its ball.
Cf (x2,y2) ≤ 4 Cf (x1,y1)
Cs (x2,y2) ≤ Cs (x1,y1)
1/4
Assign j’ completely to i*. R R j’
This will dissolve ball j’.
1/4
1/2
Now consider another client j’, being served by a facility i’, recently closed in ball of j.
Note that i’ lies in overlapping region of j and j’.

4. Questions?

5. ALGORITHM (x1,y1) is the LP Solution
is the Connection cost that client j is paying in (x1,y1) R
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
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