LP-based Algorithms for Capacitated Facility Location

Chaitanya Swamy

Joint work with Retsef Levi and David Shmoys

Cornell University

Capacitated Facility Location (CFL)

F : set of facilitiesD : set of clients.

Facility i has facility cost = fi

capacity = ui

cij : distance between points i and j

client

facility2

3

3

3

2

4

Want to:

Goal: Minimize total cost = ∑iA fi + ∑jD ci(j)j

= facility cost + client assignment cost

facility

client

23

3

3

2

4

2) Assign client j to an open facility i(j).

At most ui clients may be assigned to i.

2

1) Choose a set A of facilities to open.

open facility3

3

4

Related Work

• Korupolu, Plaxton & Rajaraman:– First constant-approx. algorithm.– Handle uniform capacities.

• Chudak & Williamson:– Simplified and improved analysis.

• Pal, Tardos & Wexler:– Constant-approximation

algorithm for non-uniform capacities.

Improvements by Mahdian & Pal, Zhang, Chen & Ye (ZCY04).Current best factor: 5.83 (ZCY04).

All results based on local search.

Approximation Algorithms

Strengthening LP relaxations

No LP-relaxation is known for CFL with a constant integrality gap.

• Padberg, Van Roy & Wolsey:– considered single-client CFL

problem– gave extended flow cover

inequalities; integrality gap = 1 for uniform capacities.

• Aardal: – adapted flow cover inequalities

to general CFL.

• Carr, Fleischer, Leung & Phillips: – gave covering inequalities for

single-client CFL; integrality gap ≤ 2.

Our Results

• Give an LP-rounding algorithm for CFL.

First LP-based approx. algorithm.Get an approx. ratio of 5 when all facility costs are equal (capacities can be different).

• Decomposition technique to divide the CFL instance into single-client CFL problems which are solved separately.

Analysis is not a client-by-client analysis.

• Decomposition technique might be useful in analyzing sophisticated LP relaxations.

LP formulation

Minimize ∑i fiyi + ∑j,i cijxij (CFL-P)

subject to∑i xij ≥ 1 j

xij ≤ yi i, j

∑j xij ≤ uiyi i, j

yi ≤ 1 i

xij,yi ≥ 0 i, j

yi : indicates if facility i is open.

xij : indicates if client j is assigned to facility i.

ui : capacity of facility i.

∑i,j wij + uiri ≤ fi + zi i

The Dual

cij

vj, wij, zi, ri ≥ 0 i, j

Maximize ∑j vj – ∑i zi (CFL-D)

subject to

vj ≤ cij + wij + ri i, j

wij

ri

vj

j

Strong Duality: Primal and dual problems have same optimum value.

Single-Client CFL

Also called single-node fixed-charge problem.

Minimize ∑i fi Yi + ∑i ciXi (SN-P)

subject to ∑i Xi ≥ D

Xi ≤ ui Yi i

Yi ≤ 1 i

Xi, Yi ≥ 0 i

Yi : indicates if facility i is open.

Xi : demand assigned to facility i.

23

4

2

3

4ci

Demand = D

Can set Yi = Xi /ui, so

(SN-P) Minimize ∑i (fi /ui+ci)Xi

subject to ∑i Xi ≥ D

Xi ≤ ui i

Xi ≥ 0 i

Lemma: Assigning demand to facilities in (fi /ui + ci) order gives optimal solution to (SN-P).

at most one facility i such that 0 < Yi < 1.

get integer solution of cost ≤ OPTSN-P + maxi fi.

Algorithm Outline

1) Clustering.

• Partition facilities in {i : yi > 0} into clusters.

• Each cluster is “centered” around a client k and denoted by Nk. Cluster Nk takes care of demand ∑j,iNk

xij.

2) Create a 1-client CFL instance for each cluster Nk.

• Open each facility iNk with yi = 1.

• Consider total demand served by facilities in Nk with yi < 1 as located at center k – get a 1-client CFL instance.

• Solve instance to decide which other facilities to open from Nk.

3) Assign clients to open facilities.

Let (x,y) : optimal primal solution

A Sample Run2 31 4

client facility with yi > 0

A Sample Run

Partition facilities in {i : yi > 0} into clusters.

2 31 4

client facility with yi > 0cluster center

A Sample Run2 31 4

client facility with yi > 0cluster center open facility

Open each facility i with yi = 1.

A Sample Run2 31 4

client facility with yi > 0cluster center open facility

Create 1-client CFL instances – movedemand to cluster centers.

A Sample Run2 31 4

client facility with yi > 0cluster center open facility

Create 1-client CFL instances by movingdemand to cluster centers.

A Sample Run2 31 4

Solve the 1-client CFL instances to decide which facilities to open.

client facility with yi > 0cluster center open facility

A Sample Run2 31 4

client facility with yi > 0open facility

Assign clients to open facilities.

Clustering

Form clusters iteratively. Clustering ensures the following properties:

Let Fj = {i: xij > 0}, wt(S) = ∑iS yi,

ctr(i) = center of i’s cluster, i.e., iNctr(i) .

j k

ictr(i)

Fj

ci,ctr(i) ≤ 3vj

• For every client j and facility iFj, ci,ctr(i) ≤ 3vj.

• Each cluster Nk has wt(Nk) ≥ ½.

∑iNk yi ≥ ½

Analysis Overview

Let y' = {0,1}-solution constructed

y'i = 1 iff facility i is open, 0 otherwise.

Will construct x' so that (x',y') is a feasible solution to (CFL-P) and bound cost of (x',y').

Use dual variables vj to bound cost:xij > 0 cij ≤ vj.

Problem: Have –zis in the dual.

But zi > 0 yi = 1. So can open these facilities and charge all of this cost to the LP; takes care of –zis.

Overview (contd.)

j

ictr(i)

Fj

• If iFj, then ci,ctr(i) ≤ 3vj,

•wt(cluster) ≥ ½

Rest is handled by the 1-client CFL solutions.

3) Cost to move demand from cluster centers back to original client locations.

2) Cost of 1-client CFL solutions.

1) Cost of opening i with yi = 1,

assigning ∑j xij units of demand to each such i.

Taking care of –zis

For each facility i with yi = 1

– Open facility i.

– For each j, assign xij fraction to i.

Lemma: Cost incurred

= ∑i:yi=1 (fi + ∑j cijxij)

= ∑j vj (∑i:yi=1 xij) – ∑i zi.

Proof: By complementary slackness.

Corollary: ∑i:yi<1 (fiyi+ ∑j cijxij) = ∑j vj (∑i:yi<1

xij).

Cost of 1-client CFL solns.

open facility with yi = 1

OPTSN-P ≤ cost of (X, Y)

= ∑iLk fiyi + ∑iLk

cik(∑j xij)

Let Lk={iNk: yi < 1}.

facility in Lk

(x,y) induces feasible solution to (SN-P).

Set Yi = yi, Xi = ∑j xij.

k

cluster Nk

Cost of integer 1-client CFL solution for Nk

≤ OPTSN-P + maxiLk fi.

Only place where we use the fact that the facility costs are all equal.

But fi = f and wt(Nk) ≥ ½, so cost

≤ ∑iLk fiyi + ∑iLk

cik(∑j xij) + 2.∑ik

fiyi.

Cost of 1-client CFL solns.

open facility with yi = 1

OPTSN-P ≤ cost of (X, Y)

= ∑iLk fiyi + ∑iLk

cik(∑j xij)

Let Lk={iNk: yi < 1}.

facility in Lk

(x,y) induces feasible solution to (SN-P).

Set Yi = yi, Xi = ∑j xij.

k

cluster Nk

So total cost of all 1-client CFL solutions

≤ ∑i:yi<1 (3fiyi + ∑j ci,ctr(i) xij) + 2.∑i:yi=1

fi .

Cost of integer 1-client CFL solution for Nk

≤ OPTSN-P + maxiLk fi.But fi = f and wt(Nk) ≥ ½, so cost

≤ ∑iLk fiyi + ∑iLk

cik(∑j xij) + 2.∑ik

fiyi.

Moving back demands

j

ictr(i)

Fj

Fix client j.

For every facility i with yi < 1, move xij demand from ctr(i) back to j.

Cost incurred for j = ∑i:yi<1 cctr(i),j xij

≤ ∑i:yi<1 (ci,ctr(i) + cij) xij

Overall cost = ∑i:yi<1 ∑j (ci,ctr(i) +

cij) xij

Putting it all together

Recall: for each iFj, ci,ctr(i) ≤ 3vj

∑i:yi<1 (fiyi + ∑j cijxij) = ∑j vj (∑i:yi<1

xij)3 components of total cost:

1) Cost of opening i such that yi = 1, and assigning xij demand of each client j to i =

∑i:yi=1 (fi + ∑j cijxij) = ∑j vj (∑i:yi=1 xij) – ∑i

zi.2) Cost of 1-client CFL solutions ≤

∑i:yi<1 (3fiyi + ∑j ci,ctr(i) xij) + 2.∑i:yi=1

fi.3) Cost of moving back demands ≤

∑i:yi<1 ∑j (ci,ctr(i) + cij) xij

Total cost ≤ ∑i:yi=1 (3fi + ∑j cijxij) +

∑i:yi<1 (3fiyi + ∑j cijxij) + ∑j 6vj (∑i:yi<1

xij) ≤ 9(∑j vj – ∑i zi ) ≤ 9.OPT.

Theorem: Get a 9-approximation algorithm for equal facility costs.Improvement

Clustering ensures that for every client j, an xij-weight of ≥ ½ is in facilities i such that ci,ctr(i) ≤ vj.

j

Fj

S∑iS xij ≥ ½

For each iS, ci,ctr(i) ≤ vj

Can do a tighter analysis to show that algorithm is a 5-approx. algorithm.

Open Questions

• LP relaxation with a constant integrality gap? Strong formulations known for the 1-client CFL problem.

– extended flow cover inequalities of Padberg, Van Roy & Wolsey

– covering inequalities of Carr, Fleischer, Leung & Phillips

Can we use these to get a strong LP relaxation for CFL?

Thank You.