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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
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
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
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
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.
  • This article discusses value symmetry
definition of the tree
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
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
  • 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
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
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
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
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
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|
    • Instead of N=∑i=1:n |Di|
ge trees and search
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
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
  • 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
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
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
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