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LP-based Algorithms for Capacitated Facility Location

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### LP-based Algorithms for Capacitated Facility Location

Chaitanya Swamy

Joint work with Retsef Levi and David Shmoys

Cornell University

Capacitated Facility Location (CFL)

facility

3

2

2

3

client

4

3

F : set of facilities

D : set of clients.

Facility i has facility cost = fi

capacity = ui

cij : distance between points i and j

- Choose a set A of facilities to open.

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

At mostuiclients may be assigned to i.

open facility

3

2

3

4

Want to:

Goal: Minimize total cost

= ∑iÎA fi + ∑jÎDci(j)j

= facility cost + client assignment cost

3

2

2

3

facility

client

4

3

Related Work

Approximation Algorithms

- 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.

- 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.

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

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.

vj

ri

wij

The Dualj

cij

Maximize ∑j vj – ∑i zi(CFL-D)

subject to

vj ≤ cij + wij + rii, j

∑i,j wij + uiri ≤ fi + zii

vj, wij, zi, ri≥ 0 i, j

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

Single-Client CFL

4

3

3

2

ci

4

2

Demand = D

Also called single-node fixed-charge problem.

Minimize ∑i fi Yi + ∑i ciXi(SN-P)

subject to ∑iXi≥ 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.

Can set Yi = Xi /ui, so

(SN-P) º Minimize ∑i (fi /ui+ci)Xi

subject to ∑iXi≥ D

Xi ≤ ui i

Xi≥ 0 i

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

Þ at most one facility i such that0 < Yi < 1.

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

Algorithm Outline

Let (x,y) : optimal primal solution

- 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,iÎNk xij.

- Create a 1-client CFL instance for each cluster Nk.
- Open each facility iÎNk with yi = 1.
- Consider total demand served by facilities in Nk with yi < 1 as located atcenter k – get a 1-client CFL instance.
- Solve instance to decide which other facilities to open from Nk.

- Assign clients to open facilities.

A Sample Run

4

3

2

1

client

facility with yi > 0

cluster center

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

A Sample Run

4

3

2

1

client

facility with yi > 0

cluster center

open facility

Open each facility i with yi = 1.

A Sample Run

4

3

2

1

client

facility with yi > 0

cluster center

open facility

Create 1-client CFL instances – move

demand to cluster centers.

A Sample Run

4

3

2

1

client

facility with yi > 0

cluster center

open facility

Create 1-client CFL instances by moving

demand to cluster centers.

A Sample Run

4

3

2

1

client

facility with yi > 0

cluster center

open facility

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

∑iÎNk yi≥ ½

ci,ctr(i) ≤ 3vj

- Each cluster Nkhas wt(Nk) ≥ ½.

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

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

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

j

k

Fj

ctr(i)

i

Form clusters iteratively. Clustering ensures the following properties:

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 > 0Þcij ≤ vj.

Problem: Have –zis in the dual.

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

- Cost of opening i with yi = 1, assigning ∑j xij units of demand to each such i.

- Cost of 1-client CFL solutions.

- Cost to move demand from cluster centers back tooriginal client locations.

j

Fj

ctr(i)

i

- If iÎFj, then ci,ctr(i) ≤ 3vj,

- wt(cluster)≥½

Rest is handled by the 1-client CFL solutions.

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).

But fi = f and wt(Nk) ≥ ½, so cost

≤ ∑iÎLk fiyi + ∑iÎLk cik(∑j xij) + 2.∑iÎNk fiyi.

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

Cost of 1-client CFL solns.open facility with yi = 1

cluster Nk

facility in Lk

k

Let Lk={iÎNk: yi < 1}.

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

Set Yi = yi, Xi = ∑j xij.

OPTSN-P≤ cost of (X, Y)

= ∑iÎLk fiyi + ∑iÎLk cik(∑j xij)

Cost of integer 1-client CFL solution for Nk

≤OPTSN-P + maxiÎLk fi.

Cost of 1-client CFL solns.

open facility with yi = 1

cluster Nk

facility in Lk

k

Let Lk={iÎNk: yi < 1}.

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

Set Yi = yi, Xi = ∑j xij.

OPTSN-P≤ cost of (X, Y)

= ∑iÎLk fiyi + ∑iÎLk cik(∑j xij)

Cost of integer 1-client CFL solution for Nk

≤OPTSN-P + maxiÎLk fi.

But fi = f and wt(Nk) ≥ ½, so cost

≤ ∑iÎLk fiyi + ∑iÎLk cik(∑j xij) + 2.∑iÎNk fiyi.

So total cost ofall 1-client CFL solutions

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

Moving back demands

j

Fj

ctr(i)

i

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

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.

Putting it all togetherRecall: for each iÎFj, ci,ctr(i) ≤ 3vj

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

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.

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

j

S

∑iÎS xij≥ ½

Fj

For each iÎS, ci,ctr(i) ≤ vj

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

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

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?

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