Algebra of equivalent instances and its applications

Algebra of equivalent

instances and its applications

Outline

• Introduction

• The origins of equivalence

• Algebra of equivalent instances

• Equivalence polyhedra

• Complex instances

The PMP

minimize

cost of

black

edges

Combinatorial Formulation

Pseudo-Boolean Formulation

pSISSJj

ijSi

c||,:

minmin

pypC

ii

mB

,}1,0{, min)(

yy

PMP

( p = 2 )

cluster 2

cluster 1

The pseudo-Boolean polynomial

8126711

14186716

221141710

10710157

C

23123

34414

12342

41231

86425

41081

24203

87477

]8428[

]6147[

]4024[

]2807[

]5137[)(

431314

421211

432322

431433

4212112,

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyyB pC y

414321421 48222733 yyyyyyyyy

||

1

0

1 1 1

0, )(

B

r Ti

ir

n

j

pm

k

k

r

kjjpC

r

rj

y

yB y

Ti

iy

monomial

term

The pseudo-Boolean polynomial

• Unique (even though the permutation matrix is not)

• Compact

– zero differences

– p-truncation

– summation of similar monomials

Some tricks with numbers

8126711

14186716

221141710

10710157

C

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12342

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41081

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]4024[

]2807[

]5137[)(

431314

421211

432322

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yyyyyyB pC y

23123

34414

12342

41231

91950

41081

24203

87477

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14136711

231142010

10715157

'C

414321421 48222733 yyyyyyyyy414321421 48222733 yyyyyyyyy

exactly the same

]9428[

]1147[

]9024[

]5807[

]0137[)(

431314

421211

432322

431433

4212112,

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyyB pC y


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14186716

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C

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'Cany similarity ?


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221141710

10710157

C

23123

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12342

41231

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41081

24203

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7139810

31591015

121172010

399185

'C

23123

44414

12332

31241

52025

42080

02225

39785

any similarity ?


8126711

14186716

221141710

10710157

C

23123

34414

12342

41231

86425

41081

24203

87477

]8428[

]6147[

]4024[

]2807[

]5137[)(

431314

421211

432322

431433

4212112,

yyyyyy

yyyyyy

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yyyyyyB pC y

414321421 48222733 yyyyyyyyy

7139810

31591015

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'C

23123

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12332

31241

52025

42080

02225

39785

]5403[

]2229[

]0027[

]2828[

]5055[)(

431313

421211

432322

431434

4212112,'

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyyB pC y

414321421 48222733 yyyyyyyyy

exactly the same

any similarity ?

Equivalence ?

8126711

14186716

221141710

10710157

C

7139810

31591015

121172010

399185

'C

3374

35117

43210

4702

D

Equivalence

Basic properties:

• Reflexivity

• Symmetry

• Transitivity

Let us call two PMP instances defined on cost matrices C

and D equivalent if and only if they are of the same size

and:

)()( ,, yy pDpC BB

Equivalence classes

• Equivalence relation induces a partition of the space of matrices into equivalence classes

• An equivalence class is closed under a certain finite set of operations

mnIR

Operations preserving equivalence • Matrix representation of a pBp

- every row i contains monomials of i-th degree (or 0);

- monomials within each column have embedded terms.

432431421414321421 4101148222733 yyyyyyyyyyyyyyyyyy

432421431

432141

421

41110

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0033

yyyyyyyyy

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yyy





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0033

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yyy

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yyy

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0272

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yyy





432421431

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421

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824

227

0033

yyyyyyyyy

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yyy

010411

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0227

00033

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414321

421

yyyyyyyyy

yyyyyy

yyy

432431421

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412

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8402

0272

00033

yyyyyyyyy

yyyyyy

yyy

00242

42253

420040

010633

0440

21544

10054

14233

02737

06748

816733

120035

ambiguous

uni que uni queuni que

Operations preserving equivalence

1) permutation of columns

2) permutation of entries within one row

preserving embedded structure of the

columns

321421

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12

45

31

54

34

yyyyyy

yyyy

yy

421321

2121

21

54

13

45

43

yyyyyy

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yy

421321

2121

12

54

13

54

43

yyyyyy

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yy

421321

2121

21

54

13

45

43

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yyyy

yy

Operations preserving equivalence

3) rearranging coefficients of similar

monomials

4) insertion/deletion of a zero column

421321

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21

54

13

45

43

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yy

054

013

045

043

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22

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yy

421321

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21

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40

45

43

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yy

yy

How to find the “smallest” matrix

within an equivalence class

8126711

14186716

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10710157

0440

21544

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How to find the “smallest” instance


Terms in the pBp can be considered as partially ordered subsets


Hasse diagram





Definitions:

chain is a sequence of embedded terms, or a path in Hasse diagram, e.g.:

antichain is a set of terms, such that no two are embedded, e.g.:

43214212110 yyyyyyyyyy

},,{ 4324121 yyyyyyy



Hasse diagramEach chain corresponds to a

column of the costs matrix

421

21

1

11

2

7

0

yyy

yy

y

9

20

7

0




Decomposition into chains is not unique !

The problem of constructing the costs matrix with minimum number of columns

is equivalent to covering the Hasse diagram with minimum number of chains





Dilworth theorem:

The minimum number

of chains is equal to the

size of a maximum

antichain

Hasse diagram

a maximum antichain



chains


432431421

434121

412

41011

842

272

0033

yyyyyyyyy

yyyyyy

yyy

0737

21148

102133

14035

polynomial in matrix form

costs matrix

Equivalence polyhedra

pCpDpC BBD ,,mn

, :IRP

A set of all instances equivalent to the one defined by C:


Take the pBp of an instance C in some matrix form:

432431421

434121

412

410011

8402

0272

00033

yyyyyyyyy

yyyyyy

yyy

1243

2334

4121

3412 A possible

permutation

matrix

Suppose, some instance defined by costs matrix D has the same permutation

matrix and is equivalent to C (for any p)

44434241

34333231

24232221

14131211

dddd

dddd

dddd

dddd

D )()( yy DC BB must hold !


432431421

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412

410011

8402

0272

00033

yyyyyyyyy

yyyyyy

yyy

1243

2334

4121

3412

44434241

34333231

24232221

14131211

dddd

dddd

dddd

dddd

D

4:

10:

11:

0:

8:

4:

2:

2:

0:

2 :

7 :

33:.

2414432

3323431

4131421

3242321

442443

133341

2232114121

43134

34443

21112

12221

34431221

ddyyy

ddyyy

ddyyy

ddyyy

ddyy

ddyy

ddddyy

ddy

ddy

ddy

ddy

ddddconstequality of coefficients


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412

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8402

0272

00033

yyyyyyyyy

yyyyyy

yyy

1243

2334

4121

3412

44434241

34333231

24232221

14131211

dddd

dddd

dddd

dddd

D

000

000

000

000

241444243444

332313334313

324222321222

413111412111

dddddd

dddddd

dddddd

dddddd

4:

10:

11:

0:

8:

4:

2:

2:

0:

2 :

7 :

33:.

2414432

3323431

4131421

3242321

442443

133341

2232114121

43134

34443

21112

12221

34431221

ddyyy

ddyyy

ddyyy

ddyyy

ddyy

ddyy

ddddyy

ddy

ddy

ddy

ddy

ddddconstequality of coefficients nonnegativity of differences


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412

410011

8402

0272

00033

yyyyyyyyy

yyyyyy

yyy

1243

2334

4121

3412

44434241

34333231

24232221

14131211

dddd

dddd

dddd

dddd

D

4:

10:

11:

0:

8:

4:

2:

2:

0:

2 :

7 :

33:.

2414432

3323431

4131421

3242321

442443

133341

2232114121

43134

34443

21112

12221

34431221

ddyyy

ddyyy

ddyyy

ddyyy

ddyy

ddyy

ddddyy

ddy

ddy

ddy

ddy

ddddconst

000

000

000

000

241444243444

332313334313

324222321222

413111412111

dddddd

dddddd

dddddd

dddddd

0,..., 4411 dd

equality of coefficients nonnegativity of differences

nonnegativity of elements


Equivalence polyhedron:

},:IR{

}:IR{ ,,,,

0AbA DDD

BBDmn

pCpDmn

pCP

)(

,,,

CPERM

pCpC PP

Set of all instances equivalent to the one defined by C:


Theorem 1

pCpC

pCpC

CPERM

,,,,

,,,,

:holds following theof one )(,any For

PP

PP


Theorem 1

pCpC

pCpC

CPERM

,,,,

,,,,

:holds following theof one )(,any For

PP

PP

Corollary

For any equivalence class there exist a natural equivalence relation that

partitions this class into equivalence subclasses

pC ,, 1PpC ,, 2

P

pC ,, 3PpC ,, 4

P

mnIR

Equivalence polyhedraProperties of

Lemma 1

}0,:IR{,, DbDD mnpC AAP

Lemma 2

t.independenlinearly are of rows All A

||)(|| )(

|| )(

: validare )(on bounds following The

TnpmBrank

Brank

rank

A

A

A

Theorem 2

|| )dim( |||| ,, BmnPBTpn pC

|B| - number of monomials

in

|T| - number of nonzero

elements in first m-p rows of

the differences matrix

)(, ypCB

Complex instances

• Definitions

Let us call instance D a reduced version of instance C ( D = red(C) )

if it satisfies the following conditions:

polytimein from obtained becan .3

)()( .2

)()( .1

CD

CD

CD

sizesize

solutionssolutions

Complexity of instance data – minimum storage capacity needed to

embed all data sufficient for solving the initial problem to optimality:

)}(:)(min{)( CDDC redsizecomp

Complex instances)}(:)(min{)( CDDC redsizecomp

Possible reductions for PMP:

• reduction of the number of rows (p-truncation)

• reduction of the number of columns (covering Hasse diagram with fewer chains)

• local aggregation of clients (adding similar terms)

• local aggregation of locations (zero differences)

Goal: construct an instance of PMP for

which all the above reductions are

inapplicable

Complex instances

ensure that any (p-truncated) column contains no pair of

equal entries

1) reduction of the number of rows (p-truncation)

2) reduction of the number of columns (covering Hasse diagram with fewer chains)

3) local aggregation of clients (adding similar terms)

4) > local aggregation of locations (zero differences) <

Complex instances

ensure that there is no pair of similar monomials in the pBp

for all k =1...m-p the sets of the first k

entries in the columns of the

permutation matrix are pairwise

different

Hasse diagram is coverable with n

internally vertex-disjoint chains

(at least up to the level of m-p)

123

234

341

412

}4,3,2,1{}4,3,2{}4,3{}4{

}4,3,2,1{}4,3,1{}4,1{}1{

}4,3,2,1{}4,2,1{}2,1{}2{

1) reduction of the number of rows (p-truncation)

2) reduction of the number of columns (covering Hasse diagram with fewer chains)

3) > local aggregation of clients (adding similar terms) <

4) local aggregation of locations (zero differences)

Complex instances1) > reduction of the number of rows (p-truncation) <

2) > reduction of the number of columns (covering Hasse diagram with fewer

chains) <

3) local aggregation of clients (adding similar terms)

4) local aggregation of locations (zero differences)

ensured by (3)

Complex instances

kmC

k

1},min{)(

1

pm

kk

mncomp C

Optimal ratio: m=n

Complex instances

• what if 2/m

mCn

n

1 m-1

these clients

can be

aggregated

Complex instances

• Experimental results

normalized number of monomials

0.00

0.20

0.40

0.60

0.80

1.00

0 20 40 60 80 100

pmed1 rw100 rmatr100

rmatr100,1/rw100 time

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 10 20 30 40 50

MBpBM MBpBMb1 Elloumi

Running times

for OR-library

and our

instances of

corresponding

size

Complex instances

• Experimental results

Conclusions

• look for easy solvable cases in equiv.

classes

• data base of previously solved instances

• generator of benchmark instances

• estimate complexity of existing benchmark

instances

Literature• Avella, P., Sforza, A.: Logical reduction tests for the p-median problem. Annals

of Operations Research, 86, 105-115 (1999)

• Avella, P., Sassano, A., Vasil'ev, I.: Computational study of large-scale p-median problems. Mathematical Programming, Ser. A, 109, 89-114 (2007)

• Boros, E., Hammer, P.L.: Pseudo-Boolean optimization. Discrete Applied Mathematics, 123, 155-225 (2002)

• Church, R.L.: BEAMR: An exact and approximate model for the p-median problem. Computers & Operations Research, 35, 417-426 (2008)

• Cornuejols, G., Nemhauser, G., Wolsey, L.A.: A canonical representation of simple plant location problems and its applications. SIAM Journal on Matrix Analysis and Applications (SIMAX), 1(3), 261-272 (1980)

• Elloumi, S.: A tighter formulation of the p-median problem. Journal of Combinatorial Optimization, 19, 69-83 (2010)

• Goldengorin, B., Krushinsky, D.: Towards an optimal mixed-Boolean LP model for the p-median problem (submitted to Annals of Operations Research)

• Goldengorin, B., Krushinsky, D., AlBdaiwi B.F.: Complexity evaluation of benchmark instances for the p-median problem (submitted to Mathematical and Computer Modelling )

• Hammer, P.L.: Plant location -- a pseudo-Boolean approach. Israel Journal

of Technology, 6, 330-332 (1968)

• Mladenovic, N., Brimberg, J., Hansen, P., Moreno-Perez, J.A.: The p-

median problem: A survey of metaheuristic approaches. European Journal

of Operational Research, 179, 927-939 (2007)

• Reese, J.: Solution Methods for the p-Median Problem: An Annotated

Bibliography. Networks 48, 125-142 (2006)

• ReVelle, C.S., Swain, R.: Central facilities location. Geographical Analysis,

2, 30-42 (1970)

Literature (contd.)

Education

Algebra of equivalent instances and its applications