Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Detecting and breaking symmetries in circle
packing
Alberto Costa
CNRS LIX, École Polytechnique, 91128 Palaiseau, France
August 26, 2010
Mini-Workshop: Exploiting Symmetry in Optimization
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Outline
1
Basic concepts
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Outline
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Outline
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
3
Computational results
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Outline
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
3
Computational results
4
Conclusions and future work
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Abstract
The performance of Branch-and-Bound algorithms is severely
impaired by the presence of symmetric optima in a given
problem. In the circle packing problem this is evident.
A method for the automatic detection of formulation
symmetries in MINLP instances is presented.
A software implementation of this method is used to
conjecture the group structure of the problem symmetries of
packing equal circles in a square.
After the proof of the conjecture, and the presentation of
some classes of symmetry breaking constraints, the
performance of spatial Branch-and-Bound on the original
problem is compared with the performances on the
reformulations that cut away symmetric optima.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Where are we?
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
3
Computational results
4
Conclusions and future work
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing reformulation
There exists different kind of formulations for the same problem
(reformulations). In this work, we are interested in the Narrowing
reformulation [Liberti; RAIRO-RO, 2009].
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing reformulation
There exists different kind of formulations for the same problem
(reformulations). In this work, we are interested in the Narrowing
reformulation [Liberti; RAIRO-RO, 2009].
Narrowing
Each global optimum of Q corresponds to a global optimum of P ,
but there could be optima of P without a corresponding optimum
in Q.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing reformulation
There exists different kind of formulations for the same problem
(reformulations). In this work, we are interested in the Narrowing
reformulation [Liberti; RAIRO-RO, 2009].
Narrowing
Each global optimum of Q corresponds to a global optimum of P ,
but there could be optima of P without a corresponding optimum
in Q.
Basically, a Narrowing reformulation eliminates some global optima
of the original problem, but at least one is kept.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Circle Packing in a Square (CPS)
Consider the following problem: Given N ∈ N and S ∈ Q+ , can N
non-overlapping circles of unit radius be arranged in a square of
side 2S?
1
xi , yi represents the coordinates of the center of the i-th circle.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Circle Packing in a Square (CPS)
Consider the following problem: Given N ∈ N and S ∈ Q+ , can N
non-overlapping circles of unit radius be arranged in a square of
side 2S?
Non-linear Non-convex formulation1
max α
s.t.
(xi − xj )2 + (yi − yj )2 ≥ 4α
∀i < j ≤ N
xi , yi ∈ [1 − S, S − 1] ∀i ∈ N
1
xi , yi represents the coordinates of the center of the i-th circle.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Circle Packing in a Square (CPS)
Consider the following problem: Given N ∈ N and S ∈ Q+ , can N
non-overlapping circles of unit radius be arranged in a square of
side 2S?
Non-linear Non-convex formulation1
max α
s.t.
(xi − xj )2 + (yi − yj )2 ≥ 4α
∀i < j ≤ N
xi , yi ∈ [1 − S, S − 1] ∀i ∈ N
For any given N, S > 1, if a global optimum (x∗ , y ∗ , α∗ ) of CPS
has α∗ ≥ 1 then the CPS instance is a YES one.
1
xi , yi represents the coordinates of the center of the i-th circle.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Example: N=9, S=3 (www.packomania.com)
Side length of square = 6
Diameter of small circles = 2
Number of small circles = 9
www.packomania.com
Alberto Costa
 E.S PECHT
02-MAR-2010
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing in CPS
Problem
CPS has a lot of symmetric global optima. Branch-and-Bound
algorithms work bad in this situation, because the BB tree is large.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing in CPS
Problem
CPS has a lot of symmetric global optima. Branch-and-Bound
algorithms work bad in this situation, because the BB tree is large.
Possible solution
Removing some of the global optima, by adding some constraints.
This is a Narrowing reformulation.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing in CPS
Problem
CPS has a lot of symmetric global optima. Branch-and-Bound
algorithms work bad in this situation, because the BB tree is large.
Possible solution
Removing some of the global optima, by adding some constraints.
This is a Narrowing reformulation.
These “symmetry-breaking” constraints (SBCs) should be obtained
automatically, only by analyzing the mathematical formulation of
the problem.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
BB trees
Figure: Original Formulation
Alberto Costa
Figure: Narrowing Reformulation
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - some definitions (1)
Consider the folowing MINLP problem P
min{f (x) | g(x) ≤ 0 ∧ x ∈ X },
where f : Rn → R, g : Rn → Rm , x ∈ Rn , and X ⊆ Rn
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - some definitions (1)
Consider the folowing MINLP problem P
min{f (x) | g(x) ≤ 0 ∧ x ∈ X },
where f : Rn → R, g : Rn → Rm , x ∈ Rn , and X ⊆ Rn
Set-wise stabilizer
Given Y ⊆ Rn , and G ≤ Sn a group of permutations, the set-wise
stabilizer of Y w.r.t. G is a subgroup H ≤ G such that HY = Y
(we represent it by stab(Y, G)).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - some definitions (2)
Optimal solutions of P
G(P ): set of global optima for the problem P .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - some definitions (2)
Optimal solutions of P
G(P ): set of global optima for the problem P .
Solution group of P
G∗P = stab(G(P ), Sn ): solution group for the problem P , that is
the largest subgroup of Sn which maps every global optimum into
another global optimum.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - G∗P : the problem
How can we find G∗P ?
Since G∗P depends on G(P ) it cannot, in general, be found before
the solution process.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - G∗P : the problem
How can we find G∗P ?
Since G∗P depends on G(P ) it cannot, in general, be found before
the solution process.
Solution: try to find subgroups of G∗P
We consider the subgroup of G∗P consisting of all variable
permutations which “fix the formulation” of P .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - G∗P : the problem
How can we find G∗P ?
Since G∗P depends on G(P ) it cannot, in general, be found before
the solution process.
Solution: try to find subgroups of G∗P
We consider the subgroup of G∗P consisting of all variable
permutations which “fix the formulation” of P .
For π ∈ Sn and σ ∈ Sm we define σP π to be the following MINLP:
min{f (πx) | σg(πx) ≤ 0 ∧ πx ∈ X },
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - G∗P : the problem
How can we find G∗P ?
Since G∗P depends on G(P ) it cannot, in general, be found before
the solution process.
Solution: try to find subgroups of G∗P
We consider the subgroup of G∗P consisting of all variable
permutations which “fix the formulation” of P .
For π ∈ Sn and σ ∈ Sm we define σP π to be the following MINLP:
min{f (πx) | σg(πx) ≤ 0 ∧ πx ∈ X },
ḠP = {π ∈ Sn | ∃σ ∈ Sm (σP π) = P }.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
ḠP : example
Suppose there is a LP problem P = min{cT x | Ax ≤ 0 ∧ x ∈ X },
where c is the vector [111111] and A is the matrix below.
Figure: Matrix A (thanks to Leo Liberti for the example).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
ḠP : example
Suppose there is a LP problem P = min{cT x | Ax ≤ 0 ∧ x ∈ X },
where c is the vector [111111] and A is the matrix below.
Figure: Matrix A (thanks to Leo Liberti for the example).
In this case, the permutation π = (1, 2)(4, 5) is in ḠP because
σAπ = A, with σ = (3, 4), and cT π = cT , since c is composed
only by 1. Thus, π fixes the formulation of P .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - ḠP : the problem
How can we find ḠP ?
Unfortunately, for general MINLPs, determining whether
∀x ∈ dom(f ) f (πx) = f (x) and ∀x ∈ dom(g) σg(πx) = g(x) is an
undecidable problem.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - ḠP : the problem
How can we find ḠP ?
Unfortunately, for general MINLPs, determining whether
∀x ∈ dom(f ) f (πx) = f (x) and ∀x ∈ dom(g) σg(πx) = g(x) is an
undecidable problem.
We represent the objective function f and the constraints gi using
some Directed Acyclic Graphs (DAGs). Comparing the DAGs, an
oracle which can says whether these DAGs represent the same
function. (See the presentation of Leo Liberti for more details).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group - ḠP : the problem
How can we find ḠP ?
Unfortunately, for general MINLPs, determining whether
∀x ∈ dom(f ) f (πx) = f (x) and ∀x ∈ dom(g) σg(πx) = g(x) is an
undecidable problem.
We represent the objective function f and the constraints gi using
some Directed Acyclic Graphs (DAGs). Comparing the DAGs, an
oracle which can says whether these DAGs represent the same
function. (See the presentation of Leo Liberti for more details).
If the answer of the oracle is “yes”, the corresponding functions are
equals,
p but the converse may not hold (i.e. the functions sin(x)
and 1 − cos2 (x) are not recognized to be equals).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group
If the oracle considers the functions a and b equals, we write a ∼
= b.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group
If the oracle considers the functions a and b equals, we write a ∼
= b.
Formulation group
For a MP P 0 defined as min{f 0 (x) | g 0 (x) ≤ 0 ∧ x ∈ X 0 }, we write
P ∼
= P 0 if: (a) P, P 0 have the same number of variables and
constraints; (b) X = X 0 ; (c) f ∼
= f 0 and ∀i ≤ m (gi ∼
= gi0 ). We are
finally in a position to define the formulation group
GP = {π ∈ Sn | ∃σ ∈ Sm (σP π ∼
= P )} of P .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group
If the oracle considers the functions a and b equals, we write a ∼
= b.
Formulation group
For a MP P 0 defined as min{f 0 (x) | g 0 (x) ≤ 0 ∧ x ∈ X 0 }, we write
P ∼
= P 0 if: (a) P, P 0 have the same number of variables and
constraints; (b) X = X 0 ; (c) f ∼
= f 0 and ∀i ≤ m (gi ∼
= gi0 ). We are
finally in a position to define the formulation group
GP = {π ∈ Sn | ∃σ ∈ Sm (σP π ∼
= P )} of P .
It is easy to show that GP ≤ ḠP ≤ G∗P [Liberti; COCOA, 2008].
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group
If the oracle considers the functions a and b equals, we write a ∼
= b.
Formulation group
For a MP P 0 defined as min{f 0 (x) | g 0 (x) ≤ 0 ∧ x ∈ X 0 }, we write
P ∼
= P 0 if: (a) P, P 0 have the same number of variables and
constraints; (b) X = X 0 ; (c) f ∼
= f 0 and ∀i ≤ m (gi ∼
= gi0 ). We are
finally in a position to define the formulation group
GP = {π ∈ Sn | ∃σ ∈ Sm (σP π ∼
= P )} of P .
It is easy to show that GP ≤ ḠP ≤ G∗P [Liberti; COCOA, 2008].
We use GP for finding some Symmetry Breaking Constraints.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Where are we?
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
3
Computational results
4
Conclusions and future work
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group for CPS
Let C2 = S2 be the cyclic group of order 2, that represents the
permutation between x and y axes, and SN be the symmetric
group of order N , that represents the permutations of the circles.
The following theorem is true
[Hansen, Costa, Liberti; ISCO, 2010].
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Formulation group for CPS
Let C2 = S2 be the cyclic group of order 2, that represents the
permutation between x and y axes, and SN be the symmetric
group of order N , that represents the permutations of the circles.
The following theorem is true
[Hansen, Costa, Liberti; ISCO, 2010].
Theorem
The formulation group of CPS is isomorphic to C2 x SN .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (1)
Let P be the CPS formulation. For all i < j ≤ N call the
constraints (xi − xj )2 + (yi − yj )2 ≥ 4α the distance constraints.
Let (x, y, α) ∈ G(P ); the following claims are easy to establish.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (1)
Let P be the CPS formulation. For all i < j ≤ N call the
constraints (xi − xj )2 + (yi − yj )2 ≥ 4α the distance constraints.
Let (x, y, α) ∈ G(P ); the following
claims are easy to establish.
Q
1 The permutation τ =
(x
, yi ) is in GP ; (hτ i ∼
= C2 ).
i
i≤N
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (1)
Let P be the CPS formulation. For all i < j ≤ N call the
constraints (xi − xj )2 + (yi − yj )2 ≥ 4α the distance constraints.
Let (x, y, α) ∈ G(P ); the following
claims are easy to establish.
Q
1 The permutation τ =
(x
, yi ) is in GP ; (hτ i ∼
= C2 ).
i
i≤N
2 For any i ≤ N − 1, the permutation σ = (x , x
i
i i+1 )(yi , yi+1 )
∼
is in GP ; notice that hσi | i < N i = SN .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (1)
Let P be the CPS formulation. For all i < j ≤ N call the
constraints (xi − xj )2 + (yi − yj )2 ≥ 4α the distance constraints.
Let (x, y, α) ∈ G(P ); the following
claims are easy to establish.
Q
1 The permutation τ =
(x
, yi ) is in GP ; (hτ i ∼
= C2 ).
i
i≤N
2 For any i ≤ N − 1, the permutation σ = (x , x
i
i i+1 )(yi , yi+1 )
∼
is in GP ; notice that hσi | i < N i = SN .
3 Any permutation moving α to one of the variables ∈
/ GP .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (1)
Let P be the CPS formulation. For all i < j ≤ N call the
constraints (xi − xj )2 + (yi − yj )2 ≥ 4α the distance constraints.
Let (x, y, α) ∈ G(P ); the following
claims are easy to establish.
Q
1 The permutation τ =
(x
, yi ) is in GP ; (hτ i ∼
= C2 ).
i
i≤N
2 For any i ≤ N − 1, the permutation σ = (x , x
i
i i+1 )(yi , yi+1 )
∼
is in GP ; notice that hσi | i < N i = SN .
3 Any permutation moving α to one of the variables ∈
/ GP .
4 If π ∈ G
P such that π(xi ) = yi for some i ≤ N then
π(xi ) = yi for all i ≤ N , as otherwise the term xi xj + yi yj
(appearing in the distance constraints) would be mapped to a
term not appearing in the problem.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (1)
Let P be the CPS formulation. For all i < j ≤ N call the
constraints (xi − xj )2 + (yi − yj )2 ≥ 4α the distance constraints.
Let (x, y, α) ∈ G(P ); the following
claims are easy to establish.
Q
1 The permutation τ =
(x
, yi ) is in GP ; (hτ i ∼
= C2 ).
i
i≤N
2 For any i ≤ N − 1, the permutation σ = (x , x
i
i i+1 )(yi , yi+1 )
∼
is in GP ; notice that hσi | i < N i = SN .
3 Any permutation moving α to one of the variables ∈
/ GP .
4 If π ∈ G
P such that π(xi ) = yi for some i ≤ N then
π(xi ) = yi for all i ≤ N , as otherwise the term xi xj + yi yj
(appearing in the distance constraints) would be mapped to a
term not appearing in the problem.
5 For any i < N , if π ∈ G
P such that π(zi ) = zi+1 for some
z ∈ {x, y}, then π(zi ) = zi+1 , ∀z ∈ {x, y}; if not the term
xi xi+1 + yi yi+1 (appearing in some of the distance const.)
would be mapped to a term not appearing in the problem.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (2)
Let K = hτ i and HN = hσi | i ≤ N − 1i. Claims (1)-(2) imply
that K, HN ≤ GP . It is easy to check that KHN = HN K; it
follows that KHN ≤ GP and hence K, HN are normal subgroups
of KHN . Since K ∩ HN = {e}, we have
KHN ∼
= K × HN ∼
= C2 × S N ≤ G P .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (2)
Let K = hτ i and HN = hσi | i ≤ N − 1i. Claims (1)-(2) imply
that K, HN ≤ GP . It is easy to check that KHN = HN K; it
follows that KHN ≤ GP and hence K, HN are normal subgroups
of KHN . Since K ∩ HN = {e}, we have
KHN ∼
= K × HN ∼
= C2 × S N ≤ G P .
Now suppose π ∈ GP with π 6= e. By Claim (3), π cannot move α
so it must map xi to yj for some i < j ≤ N ; the action i → j on
the circles indices can be decomposed into a product of
transpositions i → i + 1, . . . , j − 1 → j.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Proof (2)
Let K = hτ i and HN = hσi | i ≤ N − 1i. Claims (1)-(2) imply
that K, HN ≤ GP . It is easy to check that KHN = HN K; it
follows that KHN ≤ GP and hence K, HN are normal subgroups
of KHN . Since K ∩ HN = {e}, we have
KHN ∼
= K × HN ∼
= C2 × S N ≤ G P .
Now suppose π ∈ GP with π 6= e. By Claim (3), π cannot move α
so it must map xi to yj for some i < j ≤ N ; the action i → j on
the circles indices can be decomposed into a product of
transpositions i → i + 1, . . . , j − 1 → j. Thus, by Claim (5)
(resp. 4), π involves a certain product γ of τ and σi ’s;
furthermore, since by definition γ maps xi to yj , any permutation
in GP (including π) can be obtained as a product of these
elements γ; hence π is an element of KHN , which shows
GP ≤ KHN , implying GP ∼
= C2 × SN .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
SBC
Once GP is known, we aim to find a reformulation Q of P which
ensures that at least one symmetric optimum of P is in G(Q)
(narrowing reformulation).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
SBC
Once GP is known, we aim to find a reformulation Q of P which
ensures that at least one symmetric optimum of P is in G(Q)
(narrowing reformulation).
A set of constraints h(x) ≤ 0 are SBCs with respect to π ∈ GP if
there is y ∈ G(P ) such that h(πy) ≤ 0 and h(y) 0. Adjoining
SBCs to P yields a narrowing Q of P .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
SBC
Once GP is known, we aim to find a reformulation Q of P which
ensures that at least one symmetric optimum of P is in G(Q)
(narrowing reformulation).
A set of constraints h(x) ≤ 0 are SBCs with respect to π ∈ GP if
there is y ∈ G(P ) such that h(πy) ≤ 0 and h(y) 0. Adjoining
SBCs to P yields a narrowing Q of P .
We found 3 classes of constraints, called respectively strong, weak,
mixed constraints.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Strong SBCs for CPS
Proposition
For z ∈ {x, y},
∀i ≤ N r {1}
zi−1 ≤ zi
(1)
are SBCs with respect to any π ∈ GQ .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Strong SBCs for CPS
Proposition
For z ∈ {x, y},
∀i ≤ N r {1}
zi−1 ≤ zi
(1)
are SBCs with respect to any π ∈ GQ .
Let x ∈ G(P ); since the σi generate the symmetric group acting
on the N circles, there exists a permutation π ∈ GQ such that
(zπ(i) | i ≤ N ) are ordered as in (1).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Strong SBCs for CPS
Proposition
For z ∈ {x, y},
∀i ≤ N r {1}
zi−1 ≤ zi
(1)
are SBCs with respect to any π ∈ GQ .
Let x ∈ G(P ); since the σi generate the symmetric group acting
on the N circles, there exists a permutation π ∈ GQ such that
(zπ(i) | i ≤ N ) are ordered as in (1).
strong SBCs
xi ≤ xi+1 , ∀i ∈ {1, . . . , N − 1}
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Weak SBCs for CPS
Let z ∈ {x, y}, Ω be the set of nontrivial orbits of the action of
GP on the variables indicies, and let ω ∈ Ω. Then
∀j ∈ ω xmin ω ≤ zj are SBCs with respect to GP
[Liberti; Math. Programming, 2010].
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Weak SBCs for CPS
Let z ∈ {x, y}, Ω be the set of nontrivial orbits of the action of
GP on the variables indicies, and let ω ∈ Ω. Then
∀j ∈ ω xmin ω ≤ zj are SBCs with respect to GP
[Liberti; Math. Programming, 2010].
weak SBCs
x1 ≤ zi , ∀i ∈ {1, . . . , N } , ∀z ∈ {x, y}
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Mixed SBCs
We tried to add some other SBCs to the original formulation
[Costa, Hansen, Liberti; CTW 2010].
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Mixed SBCs
We tried to add some other SBCs to the original formulation
[Costa, Hansen, Liberti; CTW 2010].
Idea: strong constraints give some conditions only for the x
coordinates of the centres of the circles; it would be better to have
also some conditions for the y coordinates.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Mixed SBCs
We tried to add some other SBCs to the original formulation
[Costa, Hansen, Liberti; CTW 2010].
Idea: strong constraints give some conditions only for the x
coordinates of the centres of the circles; it would be better to have
also some conditions for the y coordinates.
Let L = bSc. Starting from the strong constraints,
we
Nreplace
xiL ≤ xiL+1 with y1+(i−1)L ≤ y1+iL , ∀i ∈ 1, 2, . . . , L − 1 .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Idea of the proof
We have to show that the formulation with the mixed constraints
is a narrowing of the CPS.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Idea of the proof
We have to show that the formulation with the mixed constraints
is a narrowing of the CPS.
Consider a solution compatible with the strong constraints.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Idea of the proof
We have to show that the formulation with the mixed constraints
is a narrowing of the CPS.
Consider a solution compatible with the strong constraints.
If it is also compatible with the mixed constraints OK.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Idea of the proof
We have to show that the formulation with the mixed constraints
is a narrowing of the CPS.
Consider a solution compatible with the strong constraints.
If it is also compatible with the mixed constraints OK.
If not, suppose the constraint y1+(i−1)L ≤ y1+iL is violated.
Q
We can apply the permutation σi = L−1
`=0 (i + `, i + L + `)
(where the permutation (h, k) corresponds to swapping the
circles h and k).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Idea of the proof
We have to show that the formulation with the mixed constraints
is a narrowing of the CPS.
Consider a solution compatible with the strong constraints.
If it is also compatible with the mixed constraints OK.
If not, suppose the constraint y1+(i−1)L ≤ y1+iL is violated.
Q
We can apply the permutation σi = L−1
`=0 (i + `, i + L + `)
(where the permutation (h, k) corresponds to swapping the
circles h and k).
The only constraint violated after this operation is
xiL ≤ xiL+1 , but it was removed. Therefore we obtain an
equivalent solution.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Meaning of the proof
y 6
3r
6r
7r
1r
5r
8r
2r
4r
9r
-
x
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Meaning of the proof
y 6
3r
6r
7r
1r
5r
8r
2r
4r
9r
-
x
This solution respects the strong
constraints, but not the mixed
constraints.
x1 ≤ x2 , x 2 ≤ x3 , y 1 ≤ y 4 ,
x4 ≤ x5 , x 5 ≤ x6 , y 4 ≤ y 7 ,
x7 ≤ x8 , x 8 ≤ x9
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Meaning of the proof
y 6
3r
6r
7r
y 6
6
3
7r
r
r
1r
5r
8r
2r
4r
9r
-
4
2
8r
r
r
5
1
9r
r
r
-
x
x
This solution respects the strong
constraints, but not the mixed
constraints.
x1 ≤ x2 , x 2 ≤ x3 , y 1 ≤ y 4 ,
x4 ≤ x5 , x 5 ≤ x6 , y 4 ≤ y 7 ,
x7 ≤ x8 , x 8 ≤ x9
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Meaning of the proof
y 6
3r
6r
7r
y 6
6
3
7r
r
r
1r
5r
8r
2r
4r
9r
-
4
2
8r
r
r
5
1
9r
r
r
-
x
x
This solution respects the strong
constraints, but not the mixed
constraints.
Now, after the swapping, the
solution respects the mixed
constraints.
x1 ≤ x2 , x 2 ≤ x3 , y 1 ≤ y 4 ,
x1 ≤ x2 , x2 ≤ x3 , y1 ≤ y4 ,
x4 ≤ x5 , x 5 ≤ x6 , y 4 ≤ y 7 ,
x4 ≤ x5 , x5 ≤ x6 , y4 ≤ y7 ,
x7 ≤ x8 , x 8 ≤ x9
x7 ≤ x8 , x 8 ≤ x9
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Constraints in the previous example
Side length of square = 6
Diameter of small circles = 2
Number of small circles = 9
www.packomania.com
 E.S PECHT
02-MAR-2010
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Constraints in the previous example
Side length of square = 6
Diameter of small circles = 2
Number of small circles = 9
weak constraints
x1 ≤ x2 , x1 ≤ x3 , . . . , x1 ≤ x9
x1 ≤ y1 , x1 ≤ y2 . . . , x1 ≤ y9
www.packomania.com
 E.S PECHT
02-MAR-2010
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Constraints in the previous example
Side length of square = 6
Diameter of small circles = 2
Number of small circles = 9
weak constraints
x1 ≤ x2 , x1 ≤ x3 , . . . , x1 ≤ x9
x1 ≤ y1 , x1 ≤ y2 . . . , x1 ≤ y9
strong constraints
x1 ≤ x2 , x2 ≤ x3 , x3 ≤ x4 ,
x4 ≤ x5 , x 5 ≤ x6 , x 6 ≤ x 7 ,
x7 ≤ x8 , x 8 ≤ x9
www.packomania.com
 E.S PECHT
02-MAR-2010
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Constraints in the previous example
Side length of square = 6
Diameter of small circles = 2
Number of small circles = 9
weak constraints
x1 ≤ x2 , x1 ≤ x3 , . . . , x1 ≤ x9
x1 ≤ y1 , x1 ≤ y2 . . . , x1 ≤ y9
strong constraints
x1 ≤ x2 , x2 ≤ x3 , x3 ≤ x4 ,
x4 ≤ x5 , x 5 ≤ x6 , x 6 ≤ x 7 ,
x7 ≤ x8 , x 8 ≤ x9
mixed constraints
x1 ≤ x2 , x 2 ≤ x3 , y 1 ≤ y 4 ,
x4 ≤ x5 , x 5 ≤ x6 , y 4 ≤ y 7 ,
www.packomania.com
 E.S PECHT
02-MAR-2010
Alberto Costa
x7 ≤ x8 , x 8 ≤ x9
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Where are we?
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
3
Computational results
4
Conclusions and future work
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Tests
We compare the effect of three distinct symmetry-breaking devices:
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Tests
We compare the effect of three distinct symmetry-breaking devices:
adjoining the weak SBCs;
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Tests
We compare the effect of three distinct symmetry-breaking devices:
adjoining the weak SBCs;
adjoining the strong SBCs;
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Tests
We compare the effect of three distinct symmetry-breaking devices:
adjoining the weak SBCs;
adjoining the strong SBCs;
adjoining the mixed SBCs.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Tests
We compare the effect of three distinct symmetry-breaking devices:
adjoining the weak SBCs;
adjoining the strong SBCs;
adjoining the mixed SBCs.
The tests were performed on one 2.4GHz Intel Xeon CPU of a
computer with 24 GB RAM running Linux, using the solver
Couenne [Belotti, Lee, Liberti, Margot; 2009].
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Results - strong VS mixed (time limit 10 h)
Strong constraints usually allow us to obtain the optimal solution
faster than the weak ones. But mixed constraints are better than
the strong ones, as shown in the next table (in all cases the time
limit was reached).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Results - strong VS mixed (time limit 10 h)
Strong constraints usually allow us to obtain the optimal solution
faster than the weak ones. But mixed constraints are better than
the strong ones, as shown in the next table (in all cases the time
limit was reached).
Statistics
Inst.
16
25
36
49
68
86
4
5
6
7
8
9
f∗
strong
nodes
tree
0.660
1
0
0
0
0
2381772
461224
49962
12577
4
4
642285
188835
23784
6090
1
1
f∗
mixed
nodes
tree
1
1
1
1
0.943
0.640
2795501
521487
76409
21366
1057
5
839240
222846
34825
10136
497
1
Alberto Costa
objective
function
f∗
BB nodes
closed
BB nodes
still on
tree
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Where are we?
1
Basic concepts
2
Symmetry Breaking Constraints for CPS
3
Computational results
4
Conclusions and future work
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Mixed SBCs helps identifying good optima early on the search
(unexpected - we don’t really know why).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Mixed SBCs helps identifying good optima early on the search
(unexpected - we don’t really know why).
Cons
This approach does not scale so well to big instances.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Mixed SBCs helps identifying good optima early on the search
(unexpected - we don’t really know why).
Cons
This approach does not scale so well to big instances.
The Upper Bound does not decrease (always 2).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Mixed SBCs helps identifying good optima early on the search
(unexpected - we don’t really know why).
Cons
This approach does not scale so well to big instances.
The Upper Bound does not decrease (always 2).
Future work
Try with other solvers.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Mixed SBCs helps identifying good optima early on the search
(unexpected - we don’t really know why).
Cons
This approach does not scale so well to big instances.
The Upper Bound does not decrease (always 2).
Future work
Try with other solvers.
Look for other SBCs.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Conclusion
Pros
Adjoining SBCs improve the performances.
Mixed SBCs helps identifying good optima early on the search
(unexpected - we don’t really know why).
Cons
This approach does not scale so well to big instances.
The Upper Bound does not decrease (always 2).
Future work
Try with other solvers.
Look for other SBCs.
Adjoin SBCs dynamically at each BB node.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Another problem
Suppose there is a MINLP problem P , where the domain of the
variables is {0, 1}n , and the formulation group of P is GP = Cn
(cyclic group of order n).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Another problem
Suppose there is a MINLP problem P , where the domain of the
variables is {0, 1}n , and the formulation group of P is GP = Cn
(cyclic group of order n).
Example, n = 5
Suppose that x∗ = (0, 1, 1, 0, 0) ∈ G(P ); as a consequence, also
{(0, 0, 1, 1, 0), (0, 0, 0, 1, 1), (1, 0, 0, 0, 1), (1, 1, 0, 0, 0)} are in
G(P )
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Another problem
Suppose there is a MINLP problem P , where the domain of the
variables is {0, 1}n , and the formulation group of P is GP = Cn
(cyclic group of order n).
Example, n = 5
Suppose that x∗ = (0, 1, 1, 0, 0) ∈ G(P ); as a consequence, also
{(0, 0, 1, 1, 0), (0, 0, 0, 1, 1), (1, 0, 0, 0, 1), (1, 1, 0, 0, 0)} are in
G(P )
How can we break these symmetries ?
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Possible solution
Let v = (n, n − 1, . . . , 1) ∈ Nn .
Let π = (1, . . . , n − 1, n) ∈ Cn , (Cn =< π >).
Could xv ≤ (π i x)v, ∀i ∈ N|i < n be SBCs for the problem P?
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Possible solution
Let v = (n, n − 1, . . . , 1) ∈ Nn .
Let π = (1, . . . , n − 1, n) ∈ Cn , (Cn =< π >).
Could xv ≤ (π i x)v, ∀i ∈ N|i < n be SBCs for the problem P?
In the previous example, only x∗ = (0, 0, 0, 1, 1) is a valid solution.
The constraints became respectively 3 ≤ 6, 3 ≤ 9, 3 ≤ 7, 3 ≤ 5.
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Possible solution
Let v = (n, n − 1, . . . , 1) ∈ Nn .
Let π = (1, . . . , n − 1, n) ∈ Cn , (Cn =< π >).
Could xv ≤ (π i x)v, ∀i ∈ N|i < n be SBCs for the problem P?
In the previous example, only x∗ = (0, 0, 0, 1, 1) is a valid solution.
The constraints became respectively 3 ≤ 6, 3 ≤ 9, 3 ≤ 7, 3 ≤ 5.
Question: Can we find SBCs if x ∈ Nn or x ∈ Rn ?
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing formulations with mixed SBCs - Proof (1)
Notation
L = bSc
N = {1, . . . , N }
N 0 = {1, . . . , N − 1}
N 00 = {1, L + 1, 2L + 1, . . . , (dN/Le − 2)L + 1}
S = {xi ≤ xi+1 | i ∈ N 0 }
Ai = {xh ≤ xh+1 | h ∈ N 0 r {i + L − 1}}
Ci = {yi ≤ yi+L } for all i ∈ N 00
CPS0 ≡ CPS ∪ S (CPS with strong constraints)
CPSi ≡ CPS ∪ Ai ∪ Ci , ∀i ∈ N 00
S
CPS00 ≡ CPS ∪ i∈N 00 (Ai ∪ Ci ).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing formulations with mixed SBCs - Proof (2)
Proposition
For all i ∈ N 00 , CPSi is a narrowing of CPS.
Proof.
Let i ∈ N 00 and (x̄, ȳ, ᾱ) ∈ G(CPS). For a permutation π ∈ SN we
assume π(x̄, ȳ, ᾱ) = (πx̄, π ȳ, ᾱ) where π acts on a vector in RN by
permuting the indices of its components; notice that since π is
simply a reindexing of the circles, π(x̄, ȳ, ᾱ) ∈ G(CPS).
Furthermore, since CPS0 is known to be a narrowing of CPS, we
can assume WLOG that (x̄, ȳ, ᾱ) satisfies S . If ȳi ≤ ȳi+L the
result holds, otherwise assume ȳi > ȳi+L . [continue]
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing formulations with mixed SBCs - Proof (3)
Proof.
Q
Consider the permutation σi = L−1
`=0 (i + `, i + L + `) in SN ;
σi (x̄, ȳ, ᾱ) has the following properties: (a) by the action of the
2-cycle (i, i + L) (appearing in σi when ` = 0) we have ȳi < ȳi+L ;
(b) ∀` ∈ {0, . . . , L − 2} we have
σi x̄i+` = x̄i+L+` ≤ x̄i+L+`+1 = σi x̄i+`+1 and
σi x̄i+L+` = x̄i+` ≤ x̄i+`+1 = σi x̄i+L+`+1 ; (c) ∀h ∈ N 0 such that
h 6∈ Hi = {i, . . . , i + 2L − 1} we have
σi x̄h = x̄h ≤ x̄h+1 = σi x̄h+1 because σi fixes all h 6∈ Hi . Thus
σi (x̄, ȳ, ᾱ) ∈ G(CP S) and satisfies the constraints of CPSi .
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing formulations with mixed SBCs - Proof (4)
Lemma
Let n = dN/Le − 1 and Σ = {σi | i ∈ N 00 }. Then hΣi ∼
= Sn .
Proof.
Notice N 00 = {(j − 1)L + 1 | 1 ≤ j ≤ n}, and define a map
ϕ((j − 1)/L + 1) = j, under which
ϕ(Σ) = {(1, 2), (2, 3), . . . , (n − 1, n)}. This map induces a group
homomorphism ϕ̄ : hΣi → Sn given by ϕ̄(σi ) = (ϕ(i), ϕ(i) + 1),
which can be verified to be injective and surjective.
∼ Sym(I hk ),
Similarly, ∀h < k ∈ N 00 , hΣhk i = h{σi | h ≤ i < k}i =
the symmetric group on the set I hk =Q
{ϕ(h), . . . , ϕ(k)}. Thus, for
all h, k ∈ N 00 , the permutation τhk = L−1
`=0 (h + `, k + `) can be
obtained as a certain product of the σi ’s for i ∈ ϕ−1 (I hk ).
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing formulations with mixed SBCs - Proof (5)
Theorem
CPS00 is a narrowing of CPS.
Proof.
Let (x̄, ȳ, ᾱ) ∈ G(CPS), and consider the set V of all constraints
Ci ≡ {yi ≤ yi+L } violated by (x̄, ȳ, ᾱ). Let ψ be the (invertible)
map given by ψ(Ci ) = (ϕ(i), ϕ(i) + 1); then ψ(V ) is a set of
transpositions that can be partitioned into maximal non-disjoint
subsets S hk = {(ϕ(h), ϕ(h) + 1), . . . , (ϕ(k) − 1, ϕ(k))}; let T be
hk
the set of pairs (h, k) for which
QS is in the partition of ψ(V ). It
is easy to verify that if πhk =
τh+`L,k−`L then πhk ȳ
`∈I hk
h+`L<k−`L
satisfies the constraints in ψ −1 (S hk ). [continue]
Alberto Costa
Detecting and breaking symmetries in circle packing
Outline
Basic concepts
Symmetry Breaking Constraints for CPS
Computational results
Conclusions and future work
Narrowing formulations with mixed SBCs - Proof (6)
Proof.
Furthermore, by maximality
of the S hk , the permutations πhk are
Q
disjoint. Now, if π = (h,k)∈T πhk , π(x̄, ȳ, ᾱ) is such that π ȳ
satisfies
all constraints in V and πx̄ satisfies all constraints in
S
00
i∈N 00 Ai by Prop. 35. Thus π(x̄, ȳ, ᾱ) ∈ G(CPS ).
Alberto Costa
Detecting and breaking symmetries in circle packing