Presented by: Shant Karakashian Symmetries in CP, Sprint 2010

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Tractable Symmetry Breaking Using Restricted Search Trees Colva M. Roney-Dougal , Ian P. Gent, Tom Kelsey, Steve Linton. Presented by: Shant Karakashian Symmetries in CP, Sprint 2010. Outline. Symmetry breaking approaches Group equivalence tree (GE-tree) Importance of GE-trees

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Tractable Symmetry Breaking Using Restricted Search TreesColva M. Roney-Dougal, Ian P. Gent, Tom Kelsey, Steve Linton

Presented by: ShantKarakashian

Symmetries in CP, Sprint 2010

Outline
• Symmetry breaking approaches
• Group equivalence tree (GE-tree)
• Importance of GE-trees
• Definitions:
• Definition of the tree
• Stabilizer and orbit
• Value symmetry
• Constructing GE-tree for value symmetry
• Example problem:
• GE-Tree construction
• Properties of the constructed tree
• GE-trees with:
• Global value symmetries
• Search
• Other symmetries
• Experiments
• Conclusion
Symmetry Breaking Approaches
• Different 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
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 node
• Given a full assignment A there is at least one leaf of the tree in AG
• A GE-tree is minimal if the deletion of any node (and its descendants) will delete at least one full assignment
Importance of GE-Trees
• Any search tree contains a GE-tree
• Helpful 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.
Definition of the Tree
• For a given tree:
• Nodes are labeled with variables
• Edges are labeled with variable values
• The state of a node N is the partial assignment given by the labels on the path from the root to N
Stabilizer and Orbit
• The stabilizer of a literal (X=a) is the set of all symmetries in G that map (X=a) to itself
• The 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
Value Symmetry
• Value Symmetry is any permutation g where:
• If (A1 = a)g = (A2 = b)
• Then A1 = A2
• The definition also allows the existence of symmetries g as:

(A1 = a)g = (A1 = b), (A2 = a)g = (A2 = c)

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 {ai : 1 ≤ i ≤ k}
• Select variable Ak+1

(can be different for the same level thus supports dynamic variable ordering)

• Label N with Ak+1
• Compute orbits:
• {Oj : 1 ≤ j ≤ ok+1} of G(1≤i≤k) on Dk+1
• Select a representative bj for each orbit Oj
• Create an edge from N labeled with bj
Example Problem
• Consider a 3x3 board with 3 pieces.
• Each piece is placed in a row
• The ithpiece placed on the second column is:
• pi = 2
• A solution to the problem is when all the pieces are on the same column
• The problem has the value symmetry group G
• The generator for G is: ,
• G= {a, b, c, d, e}
• a = b= c= d = e=
GE-Tree construction for the Example with Value Symmetries
• a = b = c = d = e =
• Orbits of {a, b, c, d, e} on Dp1 = {{1, 2, 3}}
• Choose representative {1}
• Label of 2nd node: (p1=1)
• Stabilizer = {a}
• Orbits of {a} on Dp2 = {{1}, {2, 3}}
• Choose representative {1, 2}
• Label of 3rd node: (p1=1, p2=1)
• Stabilizer = {a}

p1

p2

p3

p4

1

1

2

1

2

3

1

2

Properties of the Constructed Tree
• The constructed tree is a GE-tree
• The constructed GE-tree is minimal
• Complexity 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 t
• Deterministic algorithm: O~(N4+tN2)
• O~(x) = O(xlogcx), 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)

1212

Global Value Symmetries
• A 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 = |Ui=1:nDi|
GE-Trees and Search
• Consider the partial assignment at N and GN the stabilizer of N
• Let a and b be values in the domain of variable X
• Let O be the orbit of a under GN
• If 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
GE-Trees & Other Symmetries
• Let 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 found
• If a GE-tree is constructed for the group of value symmetries
• Then it is safe to use SBDD or SBDS to break all other symmetries
• The 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 complexities
Experiments
• Experiments conducted using the ECLiPSe CSP system
• The partial assignment and the current variable domain are passed to GAP:
• GAP returns a domain containing only symmetrically distinct values
• Coloring non-symmetric graphs:
• GE-tree compared to SBDD
• 7-coloring of a 12-vertex graph:
• 50.1 sec with SBDD
• 8.87 sec with GE-tree
Experiments (contd.)
• A most perfect magic square of size n x n
• A solution is one of 2n+1((n/2)!)2 symmetric equivalents
• Remodeled the problem to convert variable symmetries to value symmetries
Conclusion
• This work presented a new conceptual abstraction:
• A search tree containing a unique representation of each class of a full assignments
• Polynomial time algorithm to construct the tree in case of value symmetries