1. Fault Tolerant Facility Location
Chaitanya Swamy David Shmoys
Cornell University

2. Metric Facility Location2
facility 3
client 2
F : set of facilities.
D : set of clients.
Facility i has facility cost fi.
cij : distance between any i and j in D  F.
Client j wants to be connected to rj distinct facilities.
Theory Seminar, 04/2002

3. 1) Pick a set S of facilities to open.
We want to :
1) Pick a set S of facilities to open.
2) Assign each client j to rj open facilities.
Goal: Minimize total facility cost of S + sum of distances(service cost). 2
open
facility 3
facility 2
client
Theory Seminar, 04/2002

4. Previous Work
rj=1 :
LP rounding : Shmoys, Tardos & Aardal; Chudak & Shmoys; Sviridenko.
Primal-dual algorithms : Jain & Vazirani; Markakis, Mahdian, Saberi & Vazirani(MMSV01); Jain, Mahdian & Saberi(JMS02).
Local search : Koropulu, Plaxton & Rajaraman; Guha & Khuller; Charikar & Guha.
Best approx. - Mahdian, Ye & Zhang (MYZ02) : 1.52.
Theory Seminar, 04/2002

5. Previous Work (contd.) Non-uniform requirements, rj : Our Results:
Uniform requirements rj=r :
Markakis et al. (MMSV01) :
Non-uniform requirements, rj :
Jain & Vazirani : O(log rmax).
Guha, Meyerson & Munagala : 2.47.
Our Results:
Non-uniform rj : get a approx.
rj=r : can extend JMS02, MYZ02 to get a 1.52-approx.
Theory Seminar, 04/2002

6. LP Formulation Primal : Min. i fiyi + j,i cijxij s.t. i xij ≥ rj j
xij ≤ yi i, j
yi ≤ 1 i
xij ≥ 0, yi ≥ 0 i, j
Theory Seminar, 04/2002

7. LP Formulation (contd.)
Dual :
Max. j rjvj - i zi
s.t.
vj ≤ wij + cij i, j
j wij ≤ fi + zi i
vj ≥ 0, wij ≥ 0, zi ≥ 0 i, j
Theory Seminar, 04/2002

8. Complementary Slackness
Primal Slackness Conditions :
xij > 0  vj = wij + cij
yi > 0  j wij = fi + zi
Dual Slackness Conditions :
vj > 0  j xij = rj
wij > 0  xij = yi
zi > 0  yi = 1
Theory Seminar, 04/2002

9. 4-approximation : outline
≤ vj
≤ vj 2
view as rj copies
j(c) : cth copy
j(1), j(2)
Basic Idea : vj ‘pays’ for each cij s.t.
xij > 0.
Bound service cost for each copy of j by ρ·vj Þ total service cost ≤ ρ·Sj rjvj.
Problem : Have –zis in the dual.
But zi > 0 Þ yi = 1! So can open these facilities and charge all of this cost to the LP
Theory Seminar, 04/2002

10. The Algorithm
Phase 1
Clustering : Ensures that each copy j(c) has a nearby open facility.
Iterative algorithm.
S = { j|rj > 0}, Fj = { i|xij > 0} in ­fi order. 2 1 5
client in S
facility in some Fj 2 j
Cluster M
Start of iteration
Pick j with smallest vj.
Cluster is M Í Fj with SiÎM yi = rj.
Theory Seminar, 04/2002

11. facility removed from Fj
client in S 2
client not in S 3 X
facility in some Fj X X
facility opened from M j
Cluster M
facility removed from Fj X
Open rj cheapest facilities in M.
For k s.t. Fk Ç M ¹ f, connect rj copies to opened facilities. Decrease rk, set Fk=Fk-M.
End of iteration
Theory Seminar, 04/2002

12. Analysis : Phase 1
Solution is feasible : each j is connected to rj distinct facilities.
Lemma : Facility cost ≤ Si fiyi.
Proof : Cost of rj cheapest facilities in M ≤ rj· (avg. cost) = SiÎM fiyi. These facilities don’t get used again.
Theory Seminar, 04/2002

13. Analysis (contd.)
Lemma : For any j and c, service cost of copy j(c) ≤ 3vj.
Proof :
≤ vj
j(c)
≤ vk
≤ vk k
Cluster M
vk ≤ vj since k was chosen as cluster center.
Service cost ≤ vj + 2vk ≤ 3vj.
Theory Seminar, 04/2002

14. The Algorithm (contd.) Phase 2 : Taking care of –zis. X
Open all (unopened) i s.t. yi = 1.
i with yi = 1 X
i with yi < 1 and open j
rj = 3
For any j, if xij = yi = 1, disconnect a copy of j and connect it to i.
Theory Seminar, 04/2002

15. Analysis : Phase 2 Let L1 = { i | yi = 1 }, Lemma : Cost of phase 2 =
Lj = { i | xij = 1 } Í L1 and lj = |Lj|.
Lemma : Cost of phase 2 =
S fi + S cij = Sj ljvj – Si zi.
Proof : Each i with zi > 0 is opened. For iÎL1, all j s.t. wij > 0 are connected to it. So,
S vj = (service cost) + (fi + zi)
Þ Sj ljvj = S fi + S cij + Si zi.
iÎL1
j,iÎLj
j|iÎLj
iÎL1
j,iÎLj
Theory Seminar, 04/2002

16. cost for copies connected by phase 1
Finally …
Theorem : Total cost ≤ 4 times the optimal cost.
Proof : Total cost ≤
Si fiyi + 3·Sj (rj – lj)vj + S fi + S cij
iÎL1
j,iÎLj
facility cost of phase 1
cost of phase 2
cost for copies connected by phase 1
≤ Si fiyi + 3·Sj (rj – lj)vj + (Sj ljvj – Si zi )
≤ Si fiyi + 3·(Sj rjvj – Si zi ) ≤ 4·OPT.
Theory Seminar, 04/2002

17. A Randomized Algorithm
Idea : Open i with probability ρ·yi
Expected facility cost ≤ ρ·Si fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases.
Not quite… : no facility may be open!
Cluster facilities, open ≥ 1 facility in each cluster.
Theory Seminar, 04/2002

18. facilities opened in phases 1, 2
Phase 1 : Pruning out –zis.
Open all i s.t. yi = 1.
For each j, if xij = yi = 1, connect j to i.
Let Lj = { i | xij = 1 } and lj = |Lj|.
Cost = Sj ljvj – Si zi.
facilities opened in phases 1, 2
L’j
yi = 1 Lj
½ ≤ yi < 1
yi < ½ 10 Fj
Set L1
r’j = residual reqmt. = 6
Phase 2
Open all i s.t. ½ ≤ yi < 1.
For each j, let L’j = { i | ½ ≤ xij < 1 }.
Connect copies of j to i Î L’j.
Lose a factor of 2.
Set L2
Theory Seminar, 04/2002

19. Notation : facwt(S, j) = SiÎS xij
Phase 3
Notation : facwt(S, j) = SiÎS xij
Form clusters : Each cluster has facwt. ≥ ½.
Open facilities : Open ≥ 1 facility in a cluster – used as a backup facility. Open facility i with prob. 2yi.
Assign facilities to copies : Each copy j(c) gets a preferred set of facilities – P(j(c)) with facwt. ≥ ½ For c ¹ d, P(j(c)) Ç P(j(d)) = f.
Connect clients : Connect j(c) to the nearest i open in P(j(c)), or to a backup facility.
Theory Seminar, 04/2002

20. Initial Fj before any iterations
Clustering
After phases 1 and 2, Fj = { i | xij < ½ }. Sort these by ­cij and distribute among the r’j copies.
Cj(c) = avg. service cost of the cth copy, denote Sc Cj(c) = S cijxij by Cj.
iÎFj
client j
i Î Fj
Cj(2) 3
Cj(1)
Cj(3)
Initial Fj before any iterations
Want the following properties :
Clusters to be disjoint.
Each cluster have facwt. ≥ ½.
Each j be connected to r’j clusters.
Theory Seminar, 04/2002

21. Fj after some iterations
Iterative algorithm
S = { j | r’j > 0 }.
aj = ‘active’ copy of j, initially = 1.
i removed from Fj 4 X X X
i Î Fj,
serving j
Ĉj(3)
(aj) X X 1
aj = 3
facilities serving j
Fj after some iterations
(aj)
Ĉj(aj) = avg. distance to the first k facilities in Fj gathering facwt. ≥ ½;
say these facilities ‘serve’ j.
Will maintain Ĉj(aj) ≤ Cj(aj)
(aj)
Theory Seminar, 04/2002

22. X X X X Start of iteration Choose j in S with minimum vj + Ĉj(aj).
Form cluster M = facilities serving j .
Note : facilities are not split.
(aj)
aj = 1 4
client in S
facility in some Fj 2
j(3)
Cluster M
For k s.t. Fk Ç M ¹ f, decrease r’k, advance ak, set Fk = Fk – M.
aj = 2 3
facility removed from Fj X X X X 1
aj = 4
Cluster M
Theory Seminar, 04/2002

23. Opening Facilities Non-central facilitiesk
Non-central facilities j
open with prob. 2yi independent of other choices
Cluster M
Central facilities opened in 2 steps :
Open exactly 1 facility in M : i opened with prob. qyi. Acts as backup, denoted b(k ), for each k s.t. Fk Ç M ¹ f.
Open each i in M indep. with prob. (2-q)yi and independent of step 1.
(ak)
Theory Seminar, 04/2002

24. Distributing Facilities
Copy c gets a preferred set P(j(c)).
Preferred sets are disjoint.
Ensure facwt(P(j(c)), j) ≥ ½ for all c.
Possible to do so since each xij < ½.
P(j(2))
P(j(3))
P(j(1))
facility in Fj j
r’j = 3
Let Sj(c) = avg. dist. from j to P(j(c))
= (S cijxij)/facwt(P(j(c)), j)
Then Sc Sj(c) ≤ 2×Cj.
iÎP(j(c))
Theory Seminar, 04/2002

25. Analysis Feasibility follows from :
Facilities in phases 1, 2 not reused.
After clustering j is connected to r’j disjoint clusters Þ backups are distinct.
Preferred sets are disjoint.
So j connected to rj distinct facilities.
Theory Seminar, 04/2002

26. Facility cost Recall L1 = { i | yi = 1 }.
Phase 2 : incur a factor of 2.
Phase 3 : each i is opened with probability 2yi.
Expected facility cost ≤ 2×S fiyi.
for phases 2, 3
iÏL1
Theory Seminar, 04/2002

27. Service cost I Bounding backup cost, denoted by B : r.v.
D : event that no i in P(j(c)) is open.
Lemma : E[B|D] ≤ 2vj + Cj(c).
Proof : 2 cases
: backup = b(j(c)) B 1)
j(c)
≤ vj
≤ vk
k(d)
≤ Ĉj(d)
$ iÎM Ç Fj s.t. cik ≤ Ĉj(d)
j(c) 2)
≤ Ĉj(d) in expectation
≤ vj
k(d)
≤ vk
"iÎM Ç Fj, cik > Ĉj(d)
Also vk + Ĉj(d) ≤ vj + Ĉj(c) ≤ vj + Cj(c)
Theory Seminar, 04/2002

28. Service Cost II . Fix j, c. Let X(c) = service cost of j(c).d1 d2 dm
P(j(c)) sorted by increasing cij .
j(c)
i Î P(j(c))
Let di = cij, pi = prob. i is opened = 2yi.
B(c) = backup cost.
D(c) = event that no iÎP(j(c)) is open.
p = Pr[D(c)] = (1-p1)…(1-pm) ≤ e-1
davg = weighted avg. of the dis
= (Si pidi)/(Si pi) = Sj(c)
Theory Seminar, 04/2002

29. E[X(c)] = [p1d1 + (1-p1)p2d2 + … + (1-p1)…(1-pm-1)pmdm]
Then,
E[X(c)] = [p1d1 + (1-p1)p2d2 + …
+ (1-p1)…(1-pm-1)pmdm]
+ p×E[B(c)|D(c)]
≤ (1-p)×davg + p×[2vj + Cj(c)]
≤ (1-e-1)×Sj(c) + e-1×[2vj + Cj(c)].
Let X = Sc X(c) = service cost of j.
Sc Sj(c) ≤ 2×Cj and Sc Cj(c) ≤ 2×Cj.
Summing over all c = 1…r’j,
E[X] ≤ (1-e-1)×2Cj + e-1×(2r’jvj + Cj)
≤ 2×Cj + 2e-1×r’jvj.
Theory Seminar, 04/2002

30. Putting it all together
Phase 1 : pay the optimal LP cost
Phases 2, 3
Facility cost : twice LP facility cost
Service cost
Lose a factor of 2 for phase 2
Phase 3 cost is 2×(LP service cost)+2e-1×(dual value)
Overall cost for ≤ (2+2/e)×(LP cost)
phases 2, 3
Total cost ≤ (2+2/e)×OPT.
Theory Seminar, 04/2002

31. How to improve this?
Distribute facilities more equitably (in an expected sense) among copies - decreases prob. of ‘bad’ event.
Better analysis – maximum distance within a cluster can be bounded by 2Cj(c).
Balance phases 2 and 3.
Theory Seminar, 04/2002

32. Summary of Results
Give a approx. algorithm for non-uniform rjs. Based on LP rounding using complem. slackness.
For rj = r, extend the primal-dual algorithm of (JMS02) to get a approximation.
Fault tolerant k medians with rj = r
Primal-dual algorithm (JMS02) gives a 4-approx. using Lagrangean relaxation.
LP rounding gives a factor of 8.
Theory Seminar, 04/2002

33. Open Questions Reduce gap between rj = r, non- uniform rj.
Combinatorial algorithms for non-uniform rj : primal-dual, local-search.
Constant-factor approx. for fault tolerant k medians with non-uniform rjs.
Theory Seminar, 04/2002