Symmetry Breaking Ordering Constraints

Zeynep KiziltanDepartment of Information Science

Uppsala University, Sweden

http://www.dis.uu.se/~zeykiz

A progress report on work in collaboration withAlan Frisch, Brahim Hnich, Chris Jefferson, Ian Miguel, Toby Walsh

Motivations (1) An important class of symmetries in CP

matrices of decision variables rows/columns represent indistinguishable objects, hence symmetric

Rows and columns are subject to permutation

An n X m matrix model with row and column symmetry has n! X m! symmetries

grows super-exponentially

Too many symmetric search states

It can be very expensive to visit all the symmetric branches of a search tree

Motivations (2) Breaking symmetry is very important!

Breaking all row and column symmetries is difficult No one has an effective way of dealing with all row and

column symmetries. Symmetry breaking methods have to deal with very large

number of symmetries. The effort required could easily be exponential.

Aims

Main Goal Eliminate row and column symmetries effectively and

efficiently. Aims:

Investigate types of ordering constraints to break row and column symmetries.

Devise global constraints to easily pose and efficiently solve the ordering constraints.

Examine the effectiveness of the ordering constraint.

Disvantages of the Ordering Constraints

May conflict with the search strategy Increase in the size of the search tree

May already be implied during search No change in the size of the search tree

Theory may not meet practise Theory: posing lex ordering on the rows breaks row

symmetry Practise: posing lex ordering on the rows creates a search

tree bigger than or equal to no symmetry breaking

Advantages of the Ordering Constraints

Given a ”good” search strategy, many symmetries are eliminated effectively and efficiently.

The effort for breaking symmetry is polynomial posing solving

Significant reductions in the size of the search tree

Very practical for large matrices

Outline of Rest of Talk

Ordering Constraints Lexicographic Ordering Lexicographic Ordering Combined with

Sum Constraints Multiset Ordering Permutation Ordering

Future Work

Lexicographic Ordering

Used to order dictionaries

[A,B,C] ≤ lex [D,E,F]

A<D or (A=D and B<E ) or (A=D and B=E and C<F) or (A=D and B=E and C=F)

Lexicographic ordering is total. Enforcing the rows to be lexicographically

ordered breaks all row symmetry.

Breaking Row (Column) Symmetry

A B C

D E F

G H I[A B C] lex [D E F] lex [G H I] [G H I] lex [D E F] lex [A B C]

lexicographic ordering anti-lexicographic ordering

Breaking Row and Column Symmetries

Each symmetry class of assignments has at least one element where both the rows and the columns are lexicographically ordered But there may be no element with rows lex

ordered and columns anti-lex ordered To break row and column symmetries, we can

insist that the rows and columns are both lexicographically ordered (double-lex)

Double-lex breaks some but not necessarily all row and column symmetries

Lexicographic Ordering for a Pair of Vectors (1)

GivenA=[a1,...,an]

B=[b1,...,bn]taking values from {1,...d}

A ≤lex B ↔

(dn-1 a1 + ... + d0 an

) ≤ (dn-1 b1 + ... + d0 bn

) BC(≤) ↔ GAC(≤lex) Feasible for small n

Lexicographic Ordering for a Pair of Vectors (2)

Or decompositionA ≤lex B ↔

(a1 < b1) Or

(a1 = b1 And a2 < b2 )...(a1 = b1 And a2 = b2 ... an ≤ bn

)

And decomposition

A ≤lex B ↔

(a1 ≤ b1) And

(a1 = b1 -> a2 ≤ b2 )...(a1 = b1 And a2 = b2 ... -> an ≤ bn

)

Many solvers use FC =\ GAC

GAC schema is expensive

Design of global constraints for lexicographic orderings

≤lex

<lex

Consistency: GAC Complexity

Worst case: O(n) Amortised:0(1) Shared Variables Inferior to decompisitons and arithmetic

constraint

Global Constraints for Lexicographic Orderings

Lexicographic Ordering with Sum Constraints

Quite often we have a 0/1 matrix to model a collection of sets of fixed cardinality row (column) symmetry sum constraints on the rows (columns)

BIBDs, Steiner Systems, Rack Design, Steel Mill Design, ...

LexAndSum for a pair of vectors

≤lex

∑=S1

∑=S2

∑=S3

∑=S4

Design of global constraints for LexAndSum orderings ≤LexAndSum

<LexAndSum

0/1 variables Sums are non-ground Consistency: GAC Complexity

O(n)

Global Constraints for LexAndSum Orderings

Rows as Multisets

A multiset is a set with repetitions M = {{0, 1, 1, 2, 2, 3}}

Treat each row as a multiset

A B C

D E F

G H I[A B C] m [D E F] m [G H I] [G H I] m [D E F] m [A B C]

multiset ordering anti-multiset ordering

Multiset Ordering

Equal sized multisets M <m N iff

x=max(M), y=max(N) x<y OR (x=y AND M-{{x}} <m N-{{y}} )

R1=[1,2,3,2,3,1,1,2]

R2=[1,1,3,1,3,1,1,3]

R1 <m R2

Breaking Row (Column) Symmetry

Multiset ordering is partial.

Enforcing multiset ordering on the rows breaks some but not necessarily all row (column) symmetry

Breaking Row and Column Symmetries

Multiset ordering the rows is invariant to column permutation.

To break row and column symmetries, we can insist that the rows and columns are both multiset ordered (double-multiset)

Double-multiset breaks some but not necessarily all row and column symmetries

Lex + multiset

lex

≤m

M1

M2

M3

M4M4

M1≤m M2≤mM3≤m M4

Comparison: Lex vs Multiset

Lexicograhic ordering vs multiset ordering incomparable

double-lex vs double-multiset incomparable

double-lex vs lex+multiset incomparable

Multiset Ordering for a Pair of Vectors (1)

GivenA=[a1,...,an]

B=[b1,...,bn]

A ≤m B ↔ (na1+ ... + nan ) ≤ (nb1+ ... + nbn ) BC(≤) ↔ GAC(≤m) Feasible for small n

Given A=[a1,...,an]

B=[b1,...,bn]taking values from {1,...d}

Construct occurrence vectors via Regin’s gcc M=[m1,...,md]

N=[n1,...,nd]

where mi=occurrences(i,A) and ni=occurrences(i,B)

A ≤m B ↔ M ≤lex N GAC(≤m) → GAC(gcc) /\ GAC(≤lex)

Multiset Ordering for a Pair of Vectors (2)

Design of global constraints for multiset orderings

≤m

<m

Consistency: GAC Complexity

O(n) for m<n O(nlogn) for m>>n

Shared Variables

Global Constraints for Multiset Orderings

Permutation Ordering

GivenA=[a1,...,an]

B=[b1,...,bn]

A ≤perm B ↔

A ≤lex B1 And

A ≤lex B2 And... A ≤lex Bn!

B1...Bn! are permutations of B

Permutation ordering is neither total nor partial not reflexive anti-symmetric transitive

rowi ≤perm rowj for all i<j may eliminate solutions

row1 ≤perm rowi for all i>1 (first-row perm) does not eliminate solutions breaks some but not necessarily all row and column symmetries

first-row perm + first-column perm may eliminate solutions

Breaking Row and Column Symmetries

Comparisons

Permutation ordering > lexicographic ordering

first-row perm vs double-lex incomparable

First-row perm vs double-multiset incomparable

double-lex + first-row perm double-lex + row1 ≤perm rowi for all i>1 does not eliminate solutions breaks more symmetry than either of

them does not break all symmetries

Breaking More Row and Column Symmetries

How to pose Permutation Ordering?

Design of a global constraint for permutation ordering ≤perm

<perm

Consistency: GAC Complexity:

O(nlogn) Question: Can the 0(nlogn) multiset ordering algorithm easily

be modified to obtain permutation ordering? No!

Future Work

Multiset and lex when equal

LexAndLambda

A ≤lex B ∑ (Ai=Bi)=k

M1=M2 M1 lex M2

Future Work

write my thesis!