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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!
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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! • Predefine 1 solution to be canonical solution • Constraints satisfy only the canonical solution • Many symmetries = many lex-leader constraints
Simplifying lex-leader • Constraints can be simplified, or ‘pruned’ • First, simplify them individually • Second, simplify them as a set
Prune Individually • Remember that ⪯lex is transitive • Same variables can be cancelled out • Think of it ‘lexicographically’
Pruned Individually 1. ABCDEF ⪯lex ABCDEF 7. ABCDEF ⪯lex DEFABC 2. ABCDEF ⪯lex ACBDFE 8. ABCDEF ⪯lex DFEACB 3. ABCDEF ⪯lex BACEDF 9. ABCDEF ⪯lex EDFBAC 4. ABCDEF ⪯lex CBAFED 10. ABCDEF ⪯lex FEDCBA 5. ABCDEF ⪯lex BCAEFD 11. ABCDEF ⪯lex EFDBCA 6. ABCDEF ⪯lex CABFDE 12. ABCDEF ⪯lex FDECAB simplifies to 1. true 7. ABC ⪯lex DEF 2. BE ⪯lex CF 8. ABC ⪯lex DFE 3. AD ⪯lex BE 9. ABC ⪯lex EDF 4. AD ⪯lex CF 10. ABC ⪯lex FED 5. ABDE ⪯lex BCEF 11. ABCDE ⪯lex EFDBC 6. ABDE ⪯lex CAFD 12. ABCDE ⪯lex FDECA
Prune as a set • Remember that ⪯lex is transitive • Think of lex-leader constraints as a set • Constraints can simplify each other
Pruned As A Set 1. true 7. ABC ⪯lex DEF 2. BE ⪯lex CF 8. ABC ⪯lex DFE 3. AD ⪯lex BE 9. ABC ⪯lex EDF 4. AD ⪯lex CF 10. ABC ⪯lex FED 5. ABDE ⪯lex BCEF 11. ABCDE ⪯lex EFDBC 6. ABDE ⪯lex CAFD 12. ABCDE ⪯lex FDECA simplifies to 2. BE ⪯lex CF 9. ABC ⪯lex EDF 3. AD ⪯lex BE 10. ABC ⪯lex FED 7. ABC ⪯lex DEF 11. ABCD ⪯lex EFDB 8. ABC ⪯lex DFE 12. ABC ⪯lex FDE
Symmetry On All-diff • All-diff occurs often in problems with symmetry • Only variable symmetry + All-diff = great!
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
Graceful Graph • Variable symmetries for the problem are induced by the automorphism of the graph
K3 x P2 • Symmetries are isomorphic to earlier example • Thus, lex-leader constraints are the same
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
Lemma 10.23 Given 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.
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
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
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 !!!
Questions? All information and images taken from Handbook of Constraint Processing Chapter 10 Gent/Petrie/Puget
