A Study of 0/1 Encodings

Prosser & Selensky

A Study of 0/1 Encodings

Prosser & Selensky

You can encode a constraint satisfaction problem in a number ofdifferent, but logically equivalent ways. That is, each encodingexplores the same search space, but they take different times.

If you encode in a different constraint programming languageyou again get different relative performance for those encodings

This is bad news

Paper in a nutshell

You can encode a constraint satisfaction problem (with n variables)as a problem of finding an independent set of size n

That independent set is also maximal.

Will the maximality constraint improve performance?

Conclusion: maybe, maybe not. It depends on your toolkit!

Presentation in a nutshell

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6

)}9,8,7,3(),6,4(),5,4,2(),3,2,1{(

}9,8,7,6,5,4,3,2,1{

E

V

A Hypergraph

Encode CSP as independent set of size n

Independent set of a hypergraph G = (V,E)

- a set I of vertices such that no edge in E is totally subsumed by I

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4 5 7

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1

2 3

4 5 7

9

8

6}9,7,5,4,1{S

An Independent Set

You could add vertex 3 or vertex 8!

Encode CSP as independent set of size n

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4 5 7

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6

}9,8,7,6,5,2,1{S

The Largest Independent Set

Just so you know. There is only one for this graph

Represent a CSP with n variables as an independent set as follows

• each vertex of the graph corresponds to an assignment of a value to a variable• if n variables each of domain size m, we have n.m vertices

• there is an m-clique for each variable• therefore a variable can only take one value

• constraints are explicitly represented as nogoods• a nogood is a hyper edge

• select n vertices of the graph• corresponds to instantiation of all n variables

X + Y + Z = 8 where X, Y and Z are in {2,3}

An Example

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31

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2

3

3

2

2

ZZ

XX

XX

XX

XX

We have n.m 0/1 vertices

)},,(),,,(),,,(),,,(),,,{( 333223232322222 ZYXZYXZYXZYXZYXnogoods

A hyper edge for each nogood

)},(),,(),,{( 323232 ZZYYXXcliques

An m-clique for each variable’s domain

Give me an independent set of size n

9333

8233

8323

7223

8332

7232

7322

6222ZYX

X2

X3

Y2Y3

Z2

Z3

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Vv

ev

k size ofset t independen 4.

e hyperedge ofarity isr ce,independen v 3.

selectednot ex vert0 2.

selectedex vert1 1.

kv

r

iV

iVth

i

thi

Encode CSP as independent set of size n

Constraints 3 and 4 can be implemented in two ways• sum of variables equals some value (r or k) • the number of occurrences of 1 equals some value (r or k)

Could this make a difference?

A Maximal Independent Set

An independent set (as before), but we cannot add anelement to the set without loss of the independence property.

Note: the largest independent set is maximal … obviously

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}8,7,6,5,3,2{S

A Maximal Independent Set

There are 11 maximal independent sets of size 6

Remember, there is one largest independent set, size 7

CP encoding

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Encoding Maximality

)2()2(1

)2()2(0

54312

54312

VVVVV

VVVVV

That is, we state when a variable MUST be selected and when it MUST NOT be selected

An example, vertex 2

CP encoding

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on so and ...

)3()2(1

)3()2(0

)2()2(1

)2()2(0

21

20

987213

987213

54312

54312

321

321

VVVVVV

VVVVVV

VVVVV

VVVVV

VVV

VVV Example, vertices 1,2, and 3

More Generally

]1)([0

]1)([1

maximality ensure :Vertices

)()(

)()(

iji

iji

VeverticesVj

Vedgesei

VeverticesVj

Vedgesei

VearityV

VearityV

))((

ceindependen ensure :Edges

)(

earityVEeeverticesV

VV

i

i

kV

set of size :Graph

Actual Encoding of Maximality

)3()2(1 987213 VVVVVV

Example, vertice 3

We have a sum• implement using sumVars or• implement using occurs

We have the biconditional. This can be implemented in (at least) 3 ways

)()( (3)

)()( (2)

(1)

pqqp

qpqp

qp

Therefore 6 ways to implement maximality!

CSP Independent Set of Hypergraph

Add redundant maximality constraint

Solve

How it might go

So?

So, does maximality

help?This IS exciting.

An experiment

• Encode independence using sumVars (rather than occurs)• Encode biconditional using ifOnlyIf• Given a hypergraph find an independent set of size k

• using just independence constraint• using redundant maximality constraint

• Carry out experiments using• Choco 1.07• ILOG Solver 5.0

• Is run time reduced when we use maximality?

Does maximality help?

Choco 1.07

2590705018

44801696017

140803628016

102015

1840722018

55301642017

132103519016

184507035015

4029014

B

A

misisk

Conclusion: Use maximality!

Run time in milliseconds

Note: A’s largest indSet is size 14 B’s largest indSet is size 15 both encodings look for same thing

About 3 times faster

A & B are regular degree hypergraphs

Solver 5.0

60912518

148421817

345342216

626215

64112518

151621817

346940616

678178115

7812514

B

A

misisk

Conclusion:Avoid maximality!

About 8 times slower

The paper has other confounding results such as

• summation is faster than occurrence in Solver• occurrence is faster than summation in Choco• ((p & q) or (¬p & ¬q)) is fastest implementation of in Solver• ifOnlyIf is the fastest implementation of in Choco• maximality helps in Choco, and does not help in Solver

Confused?

So?

Why the differences, between Solver and Choco?

Read the paper

What does this mean? What lesson can I learn?

Be paranoid

… and maximality is interesting!

Any questions?