10.4.2 & 10.4.3
DESCRIPTION
10.4.2 & 10.4.3. Beau Michael Christ Symmetry in CSPs, Spring ’10. Overview. Quick Lex-leader review Simplifying Lex-leader constraints Symmetry with All-different. Lex-leader. Add symmetry-breaking ordering constraints Variable symmetries only! - PowerPoint PPT PresentationTRANSCRIPT
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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