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10.4.2 & 10.4.3Beau Michael Christ

Symmetry in CSPs, Spring ’10

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Overview

Quick Lex-leader review

Simplifying Lex-leader constraints

Symmetry with All-different

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Lex-leader

Add symmetry-breaking ordering constraints

Variable symmetries only!

Predefine 1 solution to be canonical solution

Constraints satisfy only the canonical solution

Many symmetries = many lex-leader constraints

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Simplifying lex-leader

Constraints can be simplified, or ‘pruned’

First, simplify them individually

Second, simplify them as a set

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Prune Individually

Remember that lex is ⪯ transitive

Same variables can be cancelled out

Think of it ‘lexicographically’

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Pruned Individually1. ABCDEF lex ABCDEF 7. ABCDEF lex ⪯ ⪯DEFABC2. ABCDEF lex ACBDFE 8. ABCDEF lex ⪯ ⪯DFEACB3. ABCDEF lex BACEDF 9. ABCDEF lex ⪯ ⪯EDFBAC4. ABCDEF lex CBAFED 10. ABCDEF lex ⪯ ⪯FEDCBA5. ABCDEF lex BCAEFD 11. ABCDEF lex ⪯ ⪯EFDBCA6. ABCDEF lex CABFDE 12. ABCDEF lex ⪯ ⪯FDECAB

1. true 7. ABC lex ⪯DEF2. BE lex CF 8. ABC lex ⪯ ⪯DFE3. AD lex BE 9. ABC lex ⪯ ⪯EDF4. AD lex CF 10. ABC lex ⪯ ⪯FED5. ABDE lex BCEF 11. ABCDE lex ⪯ ⪯EFDBC6. ABDE lex CAFD 12. ABCDE lex ⪯ ⪯FDECA

simplifies to

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Prune as a set

Remember that lex is ⪯ transitive

Think of lex-leader constraints as a set

Constraints can simplify each other

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Pruned As A Set1. true 7. ABC lex ⪯DEF2. BE lex CF 8. ABC lex ⪯ ⪯DFE3. AD lex BE 9. ABC lex ⪯ ⪯EDF4. AD lex CF 10. ABC lex ⪯ ⪯FED5. ABDE lex BCEF 11. ABCDE lex ⪯ ⪯EFDBC6. ABDE lex CAFD 12. ABCDE lex ⪯ ⪯FDECA

2. BE lex CF 9. ABC lex ⪯ ⪯EDF3. AD lex BE 10. ABC lex ⪯ ⪯FED7. ABC lex DEF 11. ABCD ⪯lex EFDB⪯8. ABC lex DFE 12. ABC lex ⪯ ⪯FDE

simplifies to

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Symmetry On All-diff

All-diff occurs often in problems with symmetry

Only variable symmetry + All-diff = great!

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Graceful graph

Graph with m edges is graceful if labeling f exists of its vertices such that:

0 <= f(i) <= m for each vertex i

the set of values f(i) are all-different

the set of values abs(f(i), f(j)) for every edge are all-different

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Graceful Graph

Variable symmetries for the problem are induced by the automorphism of the graph

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K3 x P2

Symmetries are isomorphic to earlier example

Thus, lex-leader constraints are the same

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Simplify

Take ABCDEF lex ACBDFE for example⪯A = A is obviously true

B = C cannot be true, because of all-diff

Thus, we use B < C instead

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Lemma 10.23Given a CSP where the variables V are subject to

an all-different constraint, and a variable symmetry group G for this CSP, then all variable

symmetries can be broken by adding the following constraints:

∀σ ∈ G, vs(σ) < vt(σ)

Note that if two permutations g and h are such that s(g) = s(h) and t(g) = t(h), then the corresponding constraints are identical.

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Pruned Constraints

Applying this logic, we get the following constraints:

A < B , A < C , A < D , A < E, A < F , B < C

Since A < B and B < C, we can further simplify to:

A < B , A < D , A < E, A < F , B < C

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Theorem 10.24

Given a CSP with n variables V, such that there exists an all-different constraint on these

variables, then all variable symmetries can be broken by at most n - 1 binary constraints

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Note

Not very interesting for our small problem

But it is possible to reduce a possible n! symmetries required to as little as n - 1 !!!

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Questions?

All information and imagestaken from

Handbook of Constraint ProcessingChapter 10

Gent/Petrie/Puget