Local Search Heuristics forFacility Location Problems

Rohit Khandekar

Dept. of Computer Science and Engineering,

Indian Institute of Technology Delhi

Joint work with: Vijay Arya, Naveen Garg, Vinayaka Pandit

Kamesh Munagala, Adam Meyerson

Departmental Seminar – p.1/43

Outline

Define the k-median problem

Simple local search algorithm

Analysis

Generalization

Departmental Seminar – p.2/43

The k-median problem

We are given n points in a metric space.

Departmental Seminar – p.3/43

The k-median problem

We are given n points in a metric space.

uv

w

d

�

w � v �

d

�

u � w ��

d

�

u � w � �

d

�

w � v �

d

�

u � v � �0 � d

�

u � u � 0 � d

�

u � v � d �

v � u �

Departmental Seminar – p.4/43

The k-median problem

We are given n points in a metric space.

Departmental Seminar – p.5/43

The k-median problem

We are given n points in a metric space.

We want to identify k “medians” such that the sum of distances of

all the points to their nearest medians is minimized.Departmental Seminar – p.6/43

The k-median problem

We are given n points in a metric space.

We want to identify k “medians” such that the sum of lengths of all

the red segments is minimized.Departmental Seminar – p.7/43

A brief bio-sketch of the k-median problem

NP-hard

Known to OR community since 60’s.

Used for locating warehouses, manufacturing plants, etc.

Used also for clustering, data mining.

Received the attention of the Approximation algorithmscommunity in early 90’s.

Various algorithms via LP-relaxation, primal-dual scheme,etc.

Departmental Seminar – p.8/43

A brief bio-sketch of the k-median problem

NP-hard

Known to OR community since 60’s.

Used for locating warehouses, manufacturing plants, etc.

Used also for clustering, data mining.

Received the attention of the Approximation algorithmscommunity in early 90’s.

Various algorithms via LP-relaxation, primal-dual scheme,etc.

Departmental Seminar – p.8/43

A brief bio-sketch of the k-median problem

NP-hard

Known to OR community since 60’s.

Used for locating warehouses, manufacturing plants, etc.

Used also for clustering, data mining.

Received the attention of the Approximation algorithmscommunity in early 90’s.

Various algorithms via LP-relaxation, primal-dual scheme,etc.

Departmental Seminar – p.8/43

A brief bio-sketch of the k-median problem

NP-hard

Known to OR community since 60’s.

Used for locating warehouses, manufacturing plants, etc.

Used also for clustering, data mining.

Received the attention of the Approximation algorithmscommunity in early 90’s.

Various algorithms via LP-relaxation, primal-dual scheme,etc.

Departmental Seminar – p.8/43

A brief bio-sketch of the k-median problem

NP-hard

Known to OR community since 60’s.

Used for locating warehouses, manufacturing plants, etc.

Used also for clustering, data mining.

Received the attention of the Approximation algorithmscommunity in early 90’s.

Various algorithms via LP-relaxation, primal-dual scheme,etc.

Departmental Seminar – p.8/43

A brief bio-sketch of the k-median problem

NP-hard

Known to OR community since 60’s.

Used for locating warehouses, manufacturing plants, etc.

Used also for clustering, data mining.

Received the attention of the Approximation algorithmscommunity in early 90’s.

Various algorithms via LP-relaxation, primal-dual scheme,etc.

Departmental Seminar – p.8/43

A local search algorithm

Start with any set of k medians.

Departmental Seminar – p.9/43

A local search algorithm

Start with any set of k medians.

Departmental Seminar – p.9/43

A local search algorithm

Identify a median and a point that is not a median.

Departmental Seminar – p.10/43

A local search algorithm

And SWAP tentatively!

Departmental Seminar – p.11/43

A local search algorithm

Perform the swap, only if the new solution is “better” (has lesscost) than the previous solution.

Stop, if there is no swap that improves the solution.

Departmental Seminar – p.12/43

A local search algorithm

Perform the swap, only if the new solution is “better” (has lesscost) than the previous solution.

Stop, if there is no swap that improves the solution.

Departmental Seminar – p.12/43

The algorithm

Algorithm Local Search.

1. S any k medians

2. While

�

s � S and s

� � S such that,cost

�

S � s �

s

���

cost�

S�

,do S S � s �

s

3. return S

Departmental Seminar – p.13/43

The algorithm

Algorithm Local Search.

1. S any k medians

2. While

�

s � S and s

� � S such that,cost

�

S � s �

s

��� �1 � ε �

cost

�

S

�

,do S S � s �

s

3. return S

Departmental Seminar – p.14/43

Main theorem

The local search algorithm described above computes asolution with cost (the sum of distances) at most 5 timesthe minimum cost.

Korupolu, Plaxton, and Rajaraman (1998) analyzed avariant in which they permitted adding, deleting, andswapping medians and got 3 5 ε approximation bytaking k 1 ε medians.

Departmental Seminar – p.15/43

Main theorem

The local search algorithm described above computes asolution with cost (the sum of distances) at most 5 timesthe minimum cost.

Korupolu, Plaxton, and Rajaraman (1998) analyzed avariant in which they permitted adding, deleting, andswapping medians and got

�3

�

5

�

ε

�

approximation bytaking k

�

1

� ε �

medians.

Departmental Seminar – p.15/43

Some notation

S j

js

S � � �

S

� � k NS

�

s

�

cost�

S� � the sum of lengths of all the red segments

�

Departmental Seminar – p.16/43

Some more notation

� �

O

� � k

NO

�o

�

O � �

o

Departmental Seminar – p.17/43

Some more notation

NO

�o

�

o

s1s2

s4 s3

Nos1

Nos4

Nos3

Nos2

Nos

NO

�

o

���

NS

�

s

�

Departmental Seminar – p.18/43

Local optimality of S

Since S is a local optimum solution,

We have,

cost S s o cost S for all s S o O

We shall add k of these inequalities (chosen carefully) toshow that,

cost S 5 cost O

> >

Departmental Seminar – p.19/43

Local optimality of S

Since S is a local optimum solution,

We have,

cost

�

S � s �

o

� �

cost

�

S

�

for all s � S � o � O �

We shall add k of these inequalities (chosen carefully) toshow that,

cost S 5 cost O

> >

Departmental Seminar – p.19/43

Local optimality of S

Since S is a local optimum solution,

We have,

cost

�

S � s �

o

� �

cost

�

S

�

for all s � S � o � O �

We shall add k of these inequalities (chosen carefully) toshow that,

cost

�

S

� �5 � cost

�

O

�

> >

Departmental Seminar – p.19/43

Capture

NO

�o

�

o

s1s2

s4 s3

Nos1

Nos4

Nos3

Nos2

We say that s � S captures o � O if

�No

s

����

NO

�

o

� �

2

�

Capture graphDepartmental Seminar – p.20/43

A mapping π

NO

�o

�

o

s1s2

s4 s3

Nos1

Nos4

Nos3

Nos2

We consider a permutation π : NO

�o

� � NO

�

o

�

that satisfies thefollowing property:

if s does not capture o then a point j � Nos should get mapped

outside Nos .

Departmental Seminar – p.21/43

A mapping π

NO

�o

�

o

s1s2

s4 s3

Nos1

Nos4

Nos3

Nos2

We consider a permutation π : NO

�o

� � NO

�

o

�

that satisfies thefollowing property:

if s does not capture o then a point j � Nos should get mapped

outside Nos .

Departmental Seminar – p.22/43

A mapping π

NO

�o

�

o

s1s2

s4 s3

Nos1

Nos4

Nos3

Nos2

i

�

l

2�

NO�

o� � � l

i l21

πDepartmental Seminar – p.23/43

Capture graph

O

S

l

!

l

2

Construct a bipartite graph G �O � S � E �

where there is an edge�

o � s �

if and only if s � S captures o � O.

Capture

Departmental Seminar – p.24/43

Swaps considered

O

S

l

!

l

2

“Why consider the swaps?”

Departmental Seminar – p.25/43

Swaps considered

O

S

l

!

l

2

“Why consider the swaps?”

Departmental Seminar – p.25/43

Properties of the swaps considered

O

S

l

!

l

2

If

"

s � o #

is considered, then s does not capture any o

� o.

Any o O is considered in exactly one swap.

Any s S is considered in at most 2 swaps.

Departmental Seminar – p.26/43

Properties of the swaps considered

O

S

l

!

l

2

If

"

s � o #

is considered, then s does not capture any o

� o.

Any o � O is considered in exactly one swap.

Any s S is considered in at most 2 swaps.

Departmental Seminar – p.26/43

Properties of the swaps considered

O

S

l

!

l

2

If

"

s � o #

is considered, then s does not capture any o

� o.

Any o � O is considered in exactly one swap.

Any s � S is considered in at most 2 swaps.

Departmental Seminar – p.26/43

Focus on a swap

$

s % o

&

os

Consider a swap

"s � o #

that is one of the k swaps defined above.

We know cost�

S � s �o

� �

cost

�

S

�

.

Departmental Seminar – p.27/43

Upper bound on cost

'

S ( s o

)

In the solution S � s �

o, each point is connected to theclosest median in S � s �

o.

cost S s o is the sum of distances of all the points totheir nearest medians.

We are going to demonstrate a possible way of connectingeach client to a median in S s o to get an upper boundon cost S s o .

Departmental Seminar – p.28/43

Upper bound on cost

'

S ( s o

)

In the solution S � s �

o, each point is connected to theclosest median in S � s �

o.

cost

�

S � s �

o

�

is the sum of distances of all the points totheir nearest medians.

We are going to demonstrate a possible way of connectingeach client to a median in S s o to get an upper boundon cost S s o .

Departmental Seminar – p.28/43

Upper bound on cost

'

S ( s o

)

In the solution S � s �

o, each point is connected to theclosest median in S � s �

o.

cost

�

S � s �

o

�

is the sum of distances of all the points totheir nearest medians.

We are going to demonstrate a possible way of connectingeach client to a median in S � s �

o to get an upper boundon cost

�

S � s �

o

�

.

Departmental Seminar – p.28/43

Upper bound on cost

'

S ( s o

)sNO

�

o

�

o

Points in NO

�

o

�

are now connected to the new median o.

Departmental Seminar – p.29/43

Upper bound on cost

'

S ( s o

)sNO

�

o

�

o

Thus, the increase in the distance for j � NO

�

o

�

is at most

O j

� S j �

Departmental Seminar – p.30/43

Upper bound on cost

'

S ( s o

)sNO

�

o

�

o

j

Consider a point j � NS

�s

�*

NO

�

o

�

.

Departmental Seminar – p.31/43

Upper bound on cost

'

S ( s o

)sNO

�

o

�

o

j

π

�

j

� s

+

Consider a point j � NS

�s

�*

NO

�

o

�

.

Suppose π�

j� � NS

�s

�

. (Note that s

� s.)

Departmental Seminar – p.32/43

Upper bound on cost

'

S ( s o

)sNO

�

o

�

o

j

π

�

j

� s

+

Consider a point j � NS

�s

�*

NO

�

o

�

.

Suppose π�

j� � NS

�s

�

. (Note that s

� s.)

Connect j to s

now.

Departmental Seminar – p.33/43

Upper bound on cost

'

S ( s o

)j

s

+π

�

j

�

Sπ

,

j

-Oπ

,

j

-

O j

o

+

New distance of j is at most O j

�

Oπ

.

j

/ �

Sπ

.

j

/ .

Therefore, the increase in the distance for j NS s NO ois at most

O j Oπ j Sπ j S j

Departmental Seminar – p.34/43

Upper bound on cost

'

S ( s o

)j

s

+π

�

j

�

Sπ

,

j

-Oπ

,

j

-

O j

o

+

New distance of j is at most O j

�

Oπ

.

j

/ �

Sπ

.

j

/ .Therefore, the increase in the distance for j � NS

�

s

� *

NO

�

o

�

is at mostO j

�

Oπ

.

j

/ �

Sπ

.

j

/ � S j �

Departmental Seminar – p.34/43

Upper bound on the increase in the cost

Lets try to count the total increase in the cost.

Points j NO o contribute at most

O j S j

Points j NS s NO o contribute at most

O j Oπ j Sπ j S j

Thus, the total increase is at most,

∑j NO o

O j S j ∑j NS s NO o

O j Oπ j Sπ j S j

Departmental Seminar – p.35/43

Upper bound on the increase in the cost

Lets try to count the total increase in the cost.

Points j � NO

�

o

�

contribute at most

�

O j

� S j

� �

Points j NS s NO o contribute at most

O j Oπ j Sπ j S j

Thus, the total increase is at most,

∑j NO o

O j S j ∑j NS s NO o

O j Oπ j Sπ j S j

Departmental Seminar – p.35/43

Upper bound on the increase in the cost

Lets try to count the total increase in the cost.

Points j � NO

�

o

�

contribute at most

�

O j

� S j

� �Points j � NS

�

s

� *

NO

�

o

�

contribute at most

�

O j

�

Oπ

.j

/ �Sπ

.j

/ � S j

� �

Thus, the total increase is at most,

∑j NO o

O j S j ∑j NS s NO o

O j Oπ j Sπ j S j

Departmental Seminar – p.35/43

Upper bound on the increase in the cost

Lets try to count the total increase in the cost.

Points j � NO

�

o

�

contribute at most

�

O j

� S j

� �Points j � NS

�

s

� *

NO

�

o

�

contribute at most

�

O j

�

Oπ

.j

/ �Sπ

.j

/ � S j

� �

Thus, the total increase is at most,

∑j 0NO

.

o

/�

O j

� S j� � ∑

j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �

Sπ

.

j

/ � S j

� �

Departmental Seminar – p.35/43

Upper bound on the increase in the cost

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.j

/ �Sπ

.j

/ � S j

�

cost S s o cost S

0

Departmental Seminar – p.36/43

Upper bound on the increase in the cost

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.j

/ �Sπ

.j

/ � S j

�

�

cost

�

S � s �

o

� � cost�

S

�

0

Departmental Seminar – p.36/43

Upper bound on the increase in the cost

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.j

/ �Sπ

.j

/ � S j

�

�

cost

�

S � s �

o

� � cost�

S

�

�0

Departmental Seminar – p.36/43

Plan

We have one such inequality for each swap

"

s � o #.

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

There are k swaps that we have defined.

O

S

l

l 2

Lets add the inequalities for all the k swaps and see whatwe get!

Departmental Seminar – p.37/43

Plan

We have one such inequality for each swap

"

s � o #.

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

There are k swaps that we have defined.

O

S

l

l 2

Lets add the inequalities for all the k swaps and see whatwe get!

Departmental Seminar – p.37/43

Plan

We have one such inequality for each swap

"

s � o #.

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

There are k swaps that we have defined.

O

S

l2

l

3

2

Lets add the inequalities for all the k swaps and see whatwe get!

Departmental Seminar – p.37/43

Plan

We have one such inequality for each swap

"

s � o #.

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

There are k swaps that we have defined.

O

S

l2

l

3

2

Lets add the inequalities for all the k swaps and see whatwe get!

Departmental Seminar – p.37/43

The first term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

Note that each o O is considered in exactly one swap.Thus, the first term added over all the swaps is

∑o O

∑j NO o

O j S j

∑j

O j S j

cost O cost S

Departmental Seminar – p.38/43

The first term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

Note that each o � O is considered in exactly one swap.

Thus, the first term added over all the swaps is

∑o O

∑j NO o

O j S j

∑j

O j S j

cost O cost S

Departmental Seminar – p.38/43

The first term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

Note that each o � O is considered in exactly one swap.Thus, the first term added over all the swaps is

∑o 0O

∑j 0NO

.o/

�O j

� S j

�

∑j

O j S j

cost O cost S

Departmental Seminar – p.38/43

The first term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

Note that each o � O is considered in exactly one swap.Thus, the first term added over all the swaps is

∑o 0O

∑j 0NO

.o/

�O j

� S j

�

∑j

�O j

� S j

�

cost O cost S

Departmental Seminar – p.38/43

The first term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/ 1

NO

.

o

/�

O j

�

Oπ

.

j

/ �Sπ

.j

/ � S j

� �

0 �

Note that each o � O is considered in exactly one swap.Thus, the first term added over all the swaps is

∑o 0O

∑j 0NO

.o/

�O j

� S j

�

∑j

�O j

� S j

�

cost

�

O

� � cost

�

S

� �

Departmental Seminar – p.38/43

The second term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

� �

0 �

Note thatO j Oπ j Sπ j S j

ThusO j Oπ j Sπ j S j 0

Thus the second term is at most

∑j NS s

O j Oπ j Sπ j S j

Departmental Seminar – p.39/43

The second term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

� �

0 �

Note thatO j

�

Oπ

.

j

/ �

Sπ

.j

/ �S j �

ThusO j Oπ j Sπ j S j 0

Thus the second term is at most

∑j NS s

O j Oπ j Sπ j S j

Departmental Seminar – p.39/43

The second term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

� �

0 �

Note thatO j

�

Oπ

.

j

/ �

Sπ

.j

/ �S j �

ThusO j

�

Oπ.

j/ �

Sπ

.j

/ � S j

�

0 �

Thus the second term is at most

∑j NS s

O j Oπ j Sπ j S j

Departmental Seminar – p.39/43

The second term 4 4 4

∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

� �

0 �

Note thatO j

�

Oπ

.

j

/ �

Sπ

.j

/ �S j �

ThusO j

�

Oπ.

j/ �

Sπ

.j

/ � S j

�

0 �

Thus the second term is at most

∑j 0NS

.s

/�

O j

�

Oπ

.

j

/ �

Sπ

.

j

/ � S j

� �

Departmental Seminar – p.39/43

The second term 4 4 4

Note that each s � S is considered in at most two swaps.

Thus, the second term added over all the swaps is at most

2 ∑s S

∑j NS s

O j Oπ j Sπ j S j

2 ∑j

O j Oπ j Sπ j S j

2 ∑j

O j ∑j

Oπ j ∑j

Sπ j ∑j

S j

4 cost O

Departmental Seminar – p.40/43

The second term 4 4 4

Note that each s � S is considered in at most two swaps.

Thus, the second term added over all the swaps is at most

2 ∑s 0S

∑j 0NS

.

s

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

�

2 ∑j

O j Oπ j Sπ j S j

2 ∑j

O j ∑j

Oπ j ∑j

Sπ j ∑j

S j

4 cost O

Departmental Seminar – p.40/43

The second term 4 4 4

Note that each s � S is considered in at most two swaps.

Thus, the second term added over all the swaps is at most

2 ∑s 0S

∑j 0NS

.

s

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

�

2 ∑j

�

O j

�

Oπ.

j/ �

Sπ

.

j

/ � S j

�

2 ∑j

O j ∑j

Oπ j ∑j

Sπ j ∑j

S j

4 cost O

Departmental Seminar – p.40/43

The second term 4 4 4

Note that each s � S is considered in at most two swaps.

Thus, the second term added over all the swaps is at most

2 ∑s 0S

∑j 0NS

.

s

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

�

2 ∑j

�

O j

�

Oπ.

j/ �

Sπ

.

j

/ � S j

�

2 ∑j

O j� ∑

jOπ

.

j

/ � ∑j

Sπ

.

j

/ � ∑j

S j

4 cost O

Departmental Seminar – p.40/43

The second term 4 4 4

Note that each s � S is considered in at most two swaps.

Thus, the second term added over all the swaps is at most

2 ∑s 0S

∑j 0NS

.

s

/�

O j

�

Oπ

.

j

/ �

Sπ.

j/ � S j

�

2 ∑j

�

O j

�

Oπ.

j/ �

Sπ

.

j

/ � S j

�

2 ∑j

O j� ∑

jOπ

.

j

/ � ∑j

Sπ

.

j

/ � ∑j

S j 4 � cost

�

O

� �

Departmental Seminar – p.40/43

Putting things together

0

� ∑5

s 6o 7 ∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j�

Oπ

.j

/ �

Sπ

.

j

/ � S j

�

cost O cost S 4 cost O

5 cost O cost S

Therefore,cost S 5 cost O

Departmental Seminar – p.41/43

Putting things together

0

� ∑5

s 6o 7 ∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j�

Oπ

.j

/ �

Sπ

.

j

/ � S j

�

� 8

cost

�

O

� � cost

�

S

�9 � 84 � cost

�

O

�9

5 cost O cost S

Therefore,cost S 5 cost O

Departmental Seminar – p.41/43

Putting things together

0

� ∑5

s 6o 7 ∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j�

Oπ

.j

/ �

Sπ

.

j

/ � S j

�

� 8

cost

�

O

� � cost

�

S

�9 � 84 � cost

�

O

�9

5 � cost

�O

� � cost

�

S

� �

Therefore,cost S 5 cost O

Departmental Seminar – p.41/43

Putting things together

0

� ∑5

s 6o 7 ∑j 0NO

.

o

/�

O j

� S j

� � ∑j 0NS

.

s

/1

NO

.

o

/�

O j�

Oπ

.j

/ �

Sπ

.

j

/ � S j

�

� 8

cost

�

O

� � cost

�

S

�9 � 84 � cost

�

O

�9

5 � cost

�O

� � cost

�

S

� �

Therefore,cost

�S

� �5 � cost

�

O

� �

Departmental Seminar – p.41/43

A tight example

22

0 0

22

0 0

�

k : 1 �

2

�

k

�

1

�

2

22

0 0

1 1 1

11

1

�

k : 1 �; ; ;

O

S

; ; ;

cost S 4 k 1 2 k 1 2 5k 3 2

cost O 0 k 1 2 k 1 2

Departmental Seminar – p.42/43

A tight example

22

0 0

22

0 0

�

k : 1 �

2

�

k

�

1

�

2

22

0 0

1 1 1

11

1

�

k : 1 �; ; ;

O

S

; ; ;

cost

�

S

� 4 � �k � 1 � �

2

� �

k

�

1

� �

2 �

5k � 3 � �

2

cost O 0 k 1 2 k 1 2

Departmental Seminar – p.42/43

A tight example

22

0 0

22

0 0

�

k : 1 �

2

�

k

�

1

�

2

22

0 0

1 1 1

11

1

�

k : 1 �; ; ;

O

S

; ; ;

cost

�

S

� 4 � �k � 1 � �

2

� �

k

�

1

� �

2 �

5k � 3 � �

2

cost

�

O

� 0 � �k

�

1

� �

2 �

k

�

1

� �

2

Departmental Seminar – p.42/43

Doing multiple swaps

Doing single swaps yields 5 approximation.

Doing p-way swaps yields 3 2 p approximation.

Departmental Seminar – p.43/43

Doing multiple swaps

Doing single swaps yields 5 approximation.

Doing p-way swaps yields

�

3

�2

�p

�approximation.

Departmental Seminar – p.43/43