Approximating Soft-Capacitated Facility Location Problem

1. Approximating Soft-Capacitated Facility Location Problem Mohammad Mahdian, MIT Yinyu Ye, Stanford Jiawei Zhang, Stanford

2. Facility Location Problem • Given • set Cof cities (or clients), • set F of facilities, • opening cost fi for i2F, and • metric connection cost c(i,j) for i2F and j2C • Find • set SµF of facilities to open, and • an assignment : C S of cities to open facilities • To minimize • The total facility cost (i2 S fi), plus • the total connection cost (j2C c(i,(i))).

3. Example 7 2 10 5 4 3 Solution 1: open both facilities Facility cost = 10+5=15 Connection cost = 2+3+5=10 Total cost = 25 Solution 2: open right facility Facility cost = 5 Connection cost = 7+3+8=18 Total cost = 23 Solution 3: open left facility Facility cost = 10 Connection cost = 2+4+5 = 11 Total cost = 21 Optimal 5 8

4. Capacitated Variants • Hard-Capacitated Facility Location • Facility i has a capacity ui which specifies the maximum number of clients that can be assigned to it. • Best Approx ratio: 7.88 by Pal et al. using local search • Soft-Capacitated Facility Location • Facility i can serve kui clients at a cost of kfi (for every k). • In other words, the opening cost of i is fidx/uie, where x is the number of clients served by i. • We give a 2-approximation algorithm. This achieves the integrality gap of the LP.

5. Previous Results

6. Simple Reduction to UFLP • We reduce the problem to linear-cost FLP, and then to UFLP. • In linear FLP, the cost of facility i is aix + bi, where x is the # of clients that it serves. • To reduce linear FLP to UFLP, just add ai to the connection costs of all clients to i.

7. Reduction to UFLP, cont’d. • Recall that in SCFLP, the opening cost of facility i is fidx/uie. • Replace this facility by a facility of cost fi(1+(x-1)/ui). • Observe that dx/uie · 1+(x-1)/ui· 2dx/uie

8. A Simple Observation • This reduction can double the facility cost, but it does not change the connection cost. • Definition: An algorithm is a (f,c)-approx algorithm for a FLP, if it finds a solution of cost at most f F*+c C*, where F* and C* are facility and connection costs of an arbitrary solution. • Similarly, we define the notion of (f,c)-approx reduction between two FLPs.

9. Simple Observation, cont’d. • Lemma. If we have a (f,c)-reduction from problem A to problem B, and a (f,c)-algorithm for problem B, then we get a (ff,cc)-algorithm for problem A. • We have a (2,1)-reduction from SCFLP to linear-FLP. • Also, the UFLP algorithm of Jain, M., & Saberi (the JMS algorithm) is a (1,2)-approximation for UFLP.

10. The missing link • Reduction from linear-FLP to UFLP is not necessarily a (1,1)-reduction, since it moves part of the facility cost to the connection cost; However, • Theorem. If we reduce an instance of the linear-FLP to UFLP and solve it using the JMS algorithm, then we get a (1,2)-approx. solution. • Proof is by looking into the factor-revealing LP of JMS algorithm. See paper for details.

11. Results • Theorem. There is a 2-approx alg for SCFLP. This achieves the integrality gap of the natural LP relaxation of SCFLP. • Other results in the paper: A different analysis for our 1.52-approx alg for UFLP, which allows us to implement this alg in quasi-linear time, using techniques developed by Thorup.

12. Conclusion • Moral of the story: Bi-factor approx algs and approx reductions for FLPs are useful. • Open Question 1. Is there a (1,1+2/e)-approximation for UFLP? • Such an algorithm would imply a 1.463-approximation for UFLP, achieving the hardness lower bound. • Open Question 2. Better hardness results for capacitated variants?