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 .

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LP-based Algorithms for Capacitated Facility Location

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Lp based algorithms for capacitated facility location

LP-based Algorithms for Capacitated Facility Location

Chaitanya Swamy

Joint work with Retsef Levi and David Shmoys

Cornell University


Capacitated facility location cfl

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


Lp based algorithms for capacitated facility location

  • 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

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.


Lp based algorithms for capacitated facility location

Strengthening LP relaxations

  • 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

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

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.


The dual

vj

ri

wij

The Dual

j

cij

Maximize ∑j vj – ∑i zi(CFL-D)

subject to

vj ≤ cij + wij + rii, j

∑i,j wij + uiri ≤ fi + zii

vj, wij, zi, ri≥ 0i, j

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


Single client cfl

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.


Lp based algorithms for capacitated facility location

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

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

A Sample Run

4

3

2

1

client

facility with yi > 0


A sample run1

A Sample Run

4

3

2

1

client

facility with yi > 0

cluster center

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


A sample run2

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 run3

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 run4

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 run5

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.


A sample run6

A Sample Run

4

3

2

1

client

facility with yi > 0

open facility

Assign clients to open facilities.


Clustering

∑iÎNk yi≥ ½

ci,ctr(i) ≤ 3vj

  • Each cluster Nkhas wt(Nk) ≥ ½.

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

Clustering

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

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.


Overview contd

  • 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.

Overview (contd.)

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 z i s

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

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 solns1

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

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


Putting it all together

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.

Putting it all together

Recall: 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.


Lp based algorithms for capacitated facility location

Improvement

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

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

Thank You.


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