Fault Tolerant Facility Location

Theory Seminar 042002

Previous Work (contd)

Uniform requirements rj=r

bullMarkakis et al (MMSV01) 1861

Non-uniform requirements rj

bullJain amp Vazirani O(log rmax)

bullGuha Meyerson amp Munagala 247

Our Results

bullNon-uniform rj get a 2076-approx

bullrj=r can extend JMS02 MYZ02 to

get a 152-approx

Theory Seminar 042002

LP Formulation

Primal

Min i fiyi + ji cijxij

st

i xij ge rj j

xij le yi i j

yi le 1 i

xij ge 0 yi ge 0 i j

Theory Seminar 042002

LP Formulation (contd)

Max j rjvj - i zi

st

vj le wij + cij i j

j wij le fi + zi i

vj ge 0 wij ge 0 zi ge 0 i j

Dual

Theory Seminar 042002

Complementary Slackness

Primal Slackness Conditions

bullxij gt 0 vj = wij + cij

bullyi gt 0 j wij = fi + zi

Dual Slackness Conditions

bullvj gt 0 j xij = rj

bullwij gt 0 xij = yi

bullzi gt 0 yi = 1

Theory Seminar 042002

4-approximation outline

Basic Idea vj lsquopaysrsquo for each cij stxij gt 0

Bound service cost for each copy of j by ρvj total service cost leρj

rjvj

Problem Have ndashzis in the dual

But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP

2

j(1) j(2)

le vj

le vj

view as rj

copiesj(c) cth copy

Theory Seminar 042002

The Algorithm

Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm

S = j|rj gt 0 Fj = i|xij gt 0 in fi order

Start of iteration

1 Pick j with smallest vj

2 Cluster is M Fj with iM yi = rj

2

51

2

j

client in Sfacility in some Fj

Cluster M

Theory Seminar 042002

0

X XX

30

2

j

3 Open rj cheapest facilities in M

4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M

End of iteration

client in S

facility in some Fj

client not in S

X facility removed from

Fj

Cluster M

facility opened from M

Theory Seminar 042002

Analysis Phase 1

Solution is feasible each j is connected to rj distinct facilities

Lemma Facility cost lei fiyi

Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again

Theory Seminar 042002

Analysis (contd)

Lemma For any j and c service cost of copy j(c) le3vj

Proof

vk le vj since k was chosen as cluster center

Service cost le vj + 2vk le 3vj

Cluster M

j(c)le vj

le vk

le vk k

Theory Seminar 042002

The Algorithm (contd)

Phase 2 Taking care of ndashzis

1 Open all (unopened) i st yi = 1

2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i

j

rj = 3

X

i with yi = 1i with yi lt

1 and open

Theory Seminar 042002

Analysis Phase 2

Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi

Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So

vj = (service cost) + (fi +

zi)

j ljvj = fi + cij + i zi

Let L1 = i | yi = 1

Lj = i | xij = 1 L1 and lj = |Lj|

iL1 jiLj

j|iLj

jiLjiL1

Theory Seminar 042002

Finally hellip

Theorem Total cost le 4 times the optimal cost

Proof Total cost le

i fiyi + 3j (rj ndash lj)vj + fi + cij

facility cost of phase 1 cost for

copies connected by

phase 1

cost of phase 2

lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i

zi )

lei fiyi + 3(j rjvj ndash i zi )

le4OPT

iL1 jiLj

Theory Seminar 042002

A Randomized Algorithm

Idea Open i with probability ρyi

Expected facility cost le ρi fiyi

Hope that each copy j(c) has a nearby facility open and service cost decreases

Not quitehellip no facility may be open

Cluster facilities open ge 1 facility in each cluster

Theory Seminar 042002

Phase 1 Pruning out ndashzis

Open all i st yi = 1

For each j if xij = yi = 1 connect j to i

Let Lj = i | xij = 1 and lj = |Lj|

Cost = j ljvj ndash i zi

Lj

Fj10

rrsquoj = residual reqmt = 6

Lrsquoj

Phase 2

Open all i st frac12 le yi lt 1

For each j let Lrsquoj = i | frac12 le xij lt 1

Connect copies of j to i Lrsquoj

Lose a factor of 2

facilities opened in

phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12

Set L1

Set L2

Theory Seminar 042002

Phase 3

Notation facwt(S j) = iS xij

1 Form clusters Each cluster has facwt ge frac12

2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi

3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =

4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility

Theory Seminar 042002

ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies

Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj

Initial Fj before any iterations

Cj(1)

Cj(2)

Cj(3)3

i Fj

client j

Want the following properties

Clusters to be disjoint

Each cluster have facwt ge frac12

Each j be connected to rrsquoj clusters

iFj

Theory Seminar 042002

Iterative algorithm

S = j | rrsquoj gt 0

aj = lsquoactiversquo copy of j initially = 1

Ĉj(aj) = avg distance to the first k

facilities in Fj gathering facwt ge frac12

say these facilities lsquoserversquo j

Will maintain Ĉj(aj) le Cj(aj)

X

X

X1

Fj after some iterations

X i removed from Fj

i Fjserving jĈj(3)

facilities serving j

aj = 3

4X

(aj)

(aj)

(aj)

Theory Seminar 042002

Start of iteration

1 Choose j in S with minimum vj + Ĉj(aj)

2 Form cluster M = facilities serving j Note facilities are not split

3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M

2j(3) Cluster M

aj = 1

4

1

XX X

aj = 4 Cluster M

aj = 2

3

client in S

facility in some Fj

X facility removed from Fj

(aj)

Theory Seminar 042002

Opening Facilities

Central facilities opened in 2 steps

1 Open exactly 1 facility in M i opened with prob qyi Acts as backup

denoted b(k ) for each k st Fk M

2 Open each i in M indep with prob (2-q)yi and independent of step 1

Non-central facilities

Cluster M

k

open with prob 2yi independent of other choices

j

(ak

)

Theory Seminar 042002

Let Sj(c) = avg dist from j to P(j(c))

= ( cijxij)facwt(P(j(c))

j)

Then c Sj(c) le 2Cj

Distributing Facilities

iP(j(c))

j

rrsquoj = 3

P(j(1))P(j(2))

P(j(3))

Copy c gets a preferred set P(j(c))Preferred sets are disjoint

Ensure facwt(P(j(c)) j) ge frac12 for all c

Possible to do so since each xij lt frac12

facility in Fj

Theory Seminar 042002

Analysis

Feasibility follows from

1 Facilities in phases 1 2 not reused

2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct

3 Preferred sets are disjoint

So j connected to rj distinct facilities

Theory Seminar 042002

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 le 2 fiyifor phases 2 3

iL1

Theory Seminar 042002

Bounding backup cost denoted by B rv

D event that no i in P(j(c)) is open

Lemma E[B|D] le 2vj + Cj(c)

Proof 2 cases

Service cost I

iM Fj st cik le Ĉj(d)

Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)

k(d)

j(c)

le vj

le Ĉj(d)le vk

B

k(d)

j(c)

le vj

le vk

iM Fj cik gt Ĉj(d)

le Ĉj(d) in expectatio

n

1)

2)

backup = b(j(c))

Theory Seminar 042002

Service Cost II

Fix j c Let X(c) = service cost of j(c)

Let di = cij pi = prob i is opened = 2yi

B(c) = backup costD(c) = event that no iP(j(c)) is

openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1

davg = weighted avg of the dis

= (i pidi)(i pi) = Sj(c)

d1

d2 dm

P(j(c)) sorted by increasing cij

j(c)

i P(j(c))

Theory Seminar 042002

Then

E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip

+ (1-p1)hellip(1-pm-1)pmdm]

+ pE[B(c)|D(c)]

le (1-p)davg + p[2vj + Cj(c)]

le (1-e-1)Sj(c) + e-1[2vj +

Cj(c)]

Let X = c X(c) = service cost of j

c Sj(c) le 2Cj and c Cj(c) le 2Cj

Summing over all c = 1helliprrsquoj

E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)

le 2Cj + 2e-1rrsquojvj

Theory Seminar 042002

Putting it all together

Phase 1 pay the optimal LP cost

Phases 2 3

bull Facility cost twice LP facility cost

bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)

Overall cost for le (2+2e)(LP cost) phases 2 3

Total cost le (2+2e)OPT

Theory Seminar 042002

How to improve this

bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event

bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)

bull Balance phases 2 and 3

Theory Seminar 042002

Summary of Results

bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness

bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation

bullFault tolerant k medians with rj = r

a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation

b LP rounding gives a factor of 8