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

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

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 = = 11 Total cost = 21 Optimal Example

Capacitated Variants Hard-Capacitated Facility Location Hard-Capacitated Facility Location –Facility i has a capacity u i 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 Soft-Capacitated Facility Location –Facility i can serve ku i clients at a cost of kf i (for every k). –In other words, the opening cost of i is f i d x/u i e, where x is the number of clients served by i. –We give a 2-approximation algorithm. This achieves the integrality gap of the LP.

Previous Results 3 Chudak & Shmoys (uniform capacities) 4 Jain & Vazirani 3.72 Arya et al. 3 Jain, M., & Saberi 2.89 M., Ye, & Zhang 2 This paper

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

Reduction to UFLP, cont’d. Recall that in SCFLP, the opening cost of facility i is f i d x/u i e. Recall that in SCFLP, the opening cost of facility i is f i d x/u i e. Replace this facility by a facility of cost f i (1+(x-1)/u i ). Replace this facility by a facility of cost f i (1+(x-1)/u i ). Observe that Observe that d x/u i e · 1+(x-1)/u i · 2 d x/u i e d x/u i e · 1+(x-1)/u i · 2 d x/u i e

A Simple Observation This reduction can double the facility cost, but it does not change the connection cost. 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. 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. Similarly, we define the notion of (  f,  c )- approx reduction between two FLPs.

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. 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. 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. Also, the UFLP algorithm of Jain, M., & Saberi (the JMS algorithm) is a (1,2)- approximation for UFLP.

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, 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. 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. Proof is by looking into the factor-revealing LP of JMS algorithm. See paper for details.

Results Theorem. There is a 2-approx alg for SCFLP. This achieves the integrality gap of the natural LP relaxation of SCFLP. 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: 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. 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.

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