Colva M RoneyDougal Ian P Gent Tom Kelsey Steve Linton Presented by Shant Karakashian Symmetries in CP Sprint 2010 Outline Symmetry breaking approaches Group equivalence tree GEtree ID: 424696
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Slide1
Tractable Symmetry Breaking Using Restricted Search TreesColva M. Roney-Dougal, Ian P. Gent, Tom Kelsey, Steve Linton
Presented by:
Shant
Karakashian
Symmetries in CP, Sprint 2010Slide2
OutlineSymmetry breaking approachesGroup equivalence tree (GE-tree)Importance of GE-treesDefinitions:
Definition of the tree
Stabilizer and orbit
Value symmetryConstructing GE-tree for value symmetryExample problem:GE-Tree constructionProperties of the constructed treeGE-trees with:Global value symmetriesSearchOther symmetriesExperimentsConclusion
2Slide3
Symmetry Breaking ApproachesDifferent approaches taken for symmetry breaking:Adding symmetry breaking constraints before search Using constraints generated dynamically during search
Checking for duplicate nodes before visiting them
Use of techniques from computational group theory
3Slide4
Group Equivalence Tree (GE-Tree)Given a CSP with symmetry group G, GE-tree is a search tree satisfying:
No node is isomorphic under
G
to any other nodeGiven a full assignment A there is at least one leaf of the tree in AGA GE-tree is minimal if the deletion of any node (and its descendants) will delete at least one full assignment4Slide5
Importance of GE-Trees Any search tree contains a GE-treeHelpful to analyze the efficacy of a symmetry breaking technique by comparing the search tree to the corresponding minimal GE-tree
Possible to construct GE-trees for many types of symmetries: variable symmetry, value symmetry, etc.
This article discusses value symmetry
5Slide6
Definition of the TreeFor a given tree:Nodes are labeled with
variables
Edges
are labeled with variable valuesThe state of a node N is the partial assignment given by the labels on the path from the root to N 6Slide7
Stabilizer and OrbitThe stabilizer of a literal (X=
a
) is the set of all symmetries in
G that map (X=a) to itselfThe orbit of literal (X=a):Denoted (X=a)
G
Is the set of all literals that can be reached from (
X
=
a
) by a symmetry in
G
7Slide8
Value Symmetry Value Symmetry is any permutation g where:If (A
1
=
a)g = (A2 = b)Then A1 = A2 The definition also allows the existence of symmetries g as:(
A
1
=
a
)
g
= (
A
1
=
b
), (
A
2
=
a
)
g
= (
A
2
=
c)
8Slide9
Constructing GE-Tree for Value Symmetries Given a node N:
State of
N
: Λ1≤i≤k(Ai = ai)Let G(1≤i≤k)
be the subgroup of
G
that stabilizes each of {
a
i
: 1 ≤
i
≤
k
}
Select variable
A
k
+1
(can be different for the same level thus supports dynamic variable ordering)
Label
N
with
A
k
+1
Compute orbits:{O
j
: 1 ≤
j ≤ ok+1} of G(1≤i≤k) on Dk+1 Select a representative bj for each orbit OjCreate an edge from N labeled with bj
9Slide10
Example ProblemConsider a 3x3 board with 3 pieces. Each piece is placed in a
row
The
i th piece placed on the second column is: pi = 2A solution to the problem is when all the pieces are on the same columnThe problem has the value symmetry group G The generator for G is: , G
= {
a, b, c, d, e
}
a
=
b
=
c
=
d
=
e
=
10
1
2
3
1
3
2
1
2
3
3
2
1
1
23321
123132
123213
123231
1
2
3
3
1
2Slide11
GE-Tree construction for the Example with Value Symmetriesa =
b
=
c = d = e = Orbits of {a, b, c, d, e} on Dp1 = {{1, 2, 3}} Choose representative {1}Label of 2nd
node: (
p
1
=1)
Stabilizer = {
a
}
Orbits of {
a
} on
D
p
2
= {{1}, {2, 3}}
Choose representative {1, 2}
Label of 3
rd
node: (
p
1
=1,
p
2
=1)Stabilizer = {a}
11
p
1
p
2
p
3
p
4
1
233211
231321
232131
232311
233121111
2
2
2
3Slide12
Properties of the Constructed TreeThe constructed tree is a GE-treeThe constructed GE-tree is minimalComplexity of the construction algorithm:
The group is acting on a set of size N=∑
i
=1:n |Di|Need to find the generator for G of size tDeterministic algorithm: O~(N4+t
N
2
)
O
~
(
x
) =
O
(
x
log
c
x
),
c
a constant
Computing the orbits of
G
on
D
is
O(t|D|)
Assume
O
(t) < O(N)Total cost at each node is no more than O~(N4)12
12
12Slide13
Global Value SymmetriesA group consists of global value symmetries if:For all variables X,
Y
, values
a, b and symmetries g:If (X = a)g = (X = b)Then (Y =
a
)
g
= (
Y
=
b
)
The construction produces minimal GE-trees
For the complexity of the construction:
Same as before but with
N
= |
U
i
=1:
n
D
i
|
Instead of N=∑
i
=1:n |D
i
|
13Slide14
GE-Trees and SearchConsider the partial assignment at N and G
N
the stabilizer of
NLet a and b be values in the domain of variable XLet O be the orbit of a under GNIf there exists b in the orbit O
such that
b
is deleted from the domain of
A
Then there are no solutions extending
N
^ (
X
=
a
)
Such values (
a
) are not extended during dynamic construction of GE-tree for value symmetries
GE-tree compared to SBDS and SBDD:
SBDS constructs a GE-tree using all symmetries of the group
SBDD constructs a GE-tree if the dominance check is applied at every node
14Slide15
GE-Trees & Other SymmetriesLet T be a GE-tree constructed for the subgroup H
of value symmetries of the symmetry group
G
If SBDS or SBDD is performed while searching T Then exactly one of each equivalence class of solutions will be foundIf a GE-tree is constructed for the group of value symmetries Then it is safe to use SBDD or SBDS to break all other symmetriesThe combined approach has lower complexity because the GE-tree construction as described for value symmetries has a polynomial complexity while SBDD and SBDS have exponential complexities15Slide16
ExperimentsExperiments conducted using the ECLiPS
e
CSP system
The partial assignment and the current variable domain are passed to GAP:GAP returns a domain containing only symmetrically distinct valuesColoring non-symmetric graphs:GE-tree compared to SBDD7-coloring of a 12-vertex graph: 50.1 sec with SBDD 8.87 sec with GE-tree 16Slide17
Experiments (contd.)A most perfect magic square of size n x nA solution is one of 2
n
+1
((n/2)!)2 symmetric equivalentsRemodeled the problem to convert variable symmetries to value symmetries17
n
#Sols
SBDD time
GE
-tree time
4
3
0.6
0.3
8
10
130.8
90.7
12
42
15,268.0
10,930.9Slide18
ConclusionThis work presented a new conceptual abstraction: A search tree containing a unique representation of each class of a full assignmentsPolynomial time algorithm to construct the tree in case of value symmetries
18Slide19
Thank you!
19