Exploiting Structure in Symmetry Detection for CNF

1
Exploiting Structure in Symmetry Detection for CNF

• Paul T. Darga, Mark H. Liffiton,
• Karem A. Sakallah, and Igor L. Markov
• The University of Michigan

2
Structure in SAT

• Human-designed artifacts possess considerable
  structure
• Manifested in instances of Boolean satisfiability
  (SAT)
• Symmetry
• Some rearrangement of the components of the
  design that preserves its structure
• Sparsity
• Most design elements are directly related to only
  a few other elements in the whole design
• We can exploit this structure to speed up SAT
  solving!

3
Symmetry Breaking

• Shatter (Aloul et. al)
• Converts CNF formula to a colored undirected
  graph (Crawford)
• Uses nauty (McKay), a graph symmetry detection
  tool, to find the symmetries present in the graph
• Converts the symmetries back into additional
  symmetry-breaking predicates (SBPs)
• Appends the SBPs to the original formula
• The resulting formula is often much faster to
  solve

??? (a'bc)(ab'c')(b'c)(a')
? (a'bc)(ab'c')(b'c)
(a'bc) (ab'c') (b'c)
(a'bc) (ab'c') (b'c)
(a,a')(b,c')(b',c)
4
Symmetry Breaking
Sym
Total
Search
Sym
Benchmark
84.4
0.45
0.07
0.38
Hole-n
39.4
1.93
1.17
0.76
Urq
88.4
43.84
5.08
38.76
GRoute
93.9
3.45
0.21
3.24
FPGARoute
99.4
26.03
0.17
25.86
ChnlRoute
82.6
13.84
2.41
11.43
XOR
74.6
31.51
8.01
23.50
2pipe

• On all but the synthetic Urquhart instances,
  symmetry detection with nauty dominates the run
  time of the Shatter flow
• Further improvements must come from improved
  symmetry detection

5
Outline

• Graph construction
• nauty "No Automorphisms, Yes?"
• Problem description
• Partition refinement
• The search tree
• saucy a new symmetry detection tool
• Algorithmic improvements exploiting structure
• Experimental results saucy run time
  insignificant compared to SAT solver
• Conclusions and future work

6
Graph Construction for CNF
? C1C2C3 (a'bc)(ab'c')(b'c)

• Two vertices and one edge for each variable
• One vertex for each clause
• One edge for each literal in each clause
• Color literals and clauses differently

C1
c
b
a'
C3
b'
c'
a
C2
7
Symmetry Detection Problem

• What precisely is a symmetry of a graph G?
• A symmetry is a permutation ? of the labels
  assigned to vertices of G such that G? G
• The set of all symmetries is denoted Aut(G)

C1
C2
C1
G? G
G? ? G
b
a'
c
b'
c'
a
c
b
a'
C3
C3
C3
a
b'
c'
c
b
a'
b'
c'
a
C2
C1
C2

• (a,b',c')(a',b,c)

• (a,a')(b,c')(b',c)(C1,C2)

8
Symmetry Detection Problem
?
?
?
?
?
?
?
?
(1,2)
2134
?
1234
(1,4,3,2)
4123
(1,3,2)
3124
(1,2)(3,4)
2143
(3,4)
1243
(1,4,2)
4132
(1,3,4,2)
3142
(1,2,3)
2314
(2,3)
1324
(1,4,3)
4213
(1,3)
3214
(1,2,3,4)
2341
(2,3,4)
1342
(1,4)
4231
(1,3,4)
3241
(1,2,4,3)
2413
(2,4,3)
1423
(1,4,2,3)
4312
(1,3)(2,4)
3412
(1,2,4)
2431
(2,4)
1432
(1,4)(2,3)
4321
(1,3,2,4)
3421

• Problem there are n! possible labelings!
• Can we prune the search space?

9
Partition Refinement

• We can rule out many candidate labelings
• Distinguish vertices that cannot possibly be
  mapped into each other by any symmetry
• Fast distinguishing method degree

• Select a color in the graph
• Compute the number of connections every vertex
  has to that color
• Distinguish vertices within colors based on that
  count
• Repeat until coloring stabilizes
• Refinement distinguished all vertices! This
  graph has no symmetry besides the identity.

1
1
0
0
0
0
0
0
0
0
1
0
1
0
4
4
1
1
0
1
6
7
3
0
0
4
2
0
1
0
0
2
3
0
1
2
0
1
0
1
3
5
10
Partition Refinement

• In a stable coloring
• vertices in different colors definitely cannot
  map into each other in some symmetry of the graph
• vertices in the same color may map into each
  other (i.e. refinement is only an approximation)

C1
c
b
a'
C3
b'
c'
a
C2
11
The Search Tree
? 035421
?
?0 ? ?1 (2,4)(3,5) ?2 (0,3)(1,2) ?3
(0,3,5)(1,2,4) ?4 (0,5,3)(1,4,2) ?5 (0,5)(1,4)

• Out of 6! 720 possible labelings, partition
  refinement pruned away all but the six symmetries
  of the graph

?1 (2,4)(3,5)
?3 (0,3,5)(1,2,4)
?0 ?
?5 (0,5)(1,4)
?2 (0,3)(1,2)
?4 (0,5,3)(1,4,2)

• Select a non-singleton color T (for target) and
  generate T colorings, each with one element of
  T artificially distinguished from the remainder
  of T
• Discrete colorings (leaf nodes) yield likely
  symmetries

12
Pruning Using Generators

• Too many symmetries Aut(G) is O(n!)
• Group theory provides the answer generators
• Irredundant set H ? Aut(G) implicitly
  representsentire set of symmetries
• Exponential compression H ? log2Aut(G)
• We prune away subtrees guaranteed to yield
  symmetries that we can already generate with
  previously discovered symmetries

13
saucy Exploiting Structure

• nauty works very well on small graphs (and thus
  small formulas) but fails to scale
• Takes considerably longer than the SAT solver
  after adding SBPs to the CNF formula
• Runs out of memory on formulas with corresponding
  graphs having gt50,000 vertices
• saucy improvement 1 sparse representation
• saucy improvement 2 use knowledge of graph
  construction
• Clause vertices only connected to their literals
• Never connected to each other

14
saucy Algorithmic Improvements

• nauty
• Iterate over all colors
• For each vertex, count connections to refining
  color, and sort
• saucy improvement 3
• Determine directly connected colors
• For each vertex, count connections to refining
  color, and sort
• saucy improvement 4
• For each vertex in refining color, count
  connections
• For every color touched, sort the counts

15
saucy Asymptotic Performance

• Partition refinement
• nauty implementation O(n3)
• saucy improvement 4 O(n2 log n)
• Search tree size
• Worst case exponential
• No bad leaves O(n3)
• In practice O(n)
• Complete algorithm
• Worst case exponential
• No bad leaves O(n5 log n)
• Much lower in practice

16
saucy Empirical Performance
17
saucy Empirical Performance
18
Conclusions and Future Work

• CNF formulas from EDA applications exhibit
  considerable structure (symmetry and sparsity)
• saucy, a new implementation of the nauty
  symmetry-detection system
• Exploits structure to improve symmetry detection
  performance by several orders of magnitude
• Symmetry-detection time insignificant compared to
  SAT solver
• Future work
• Apply saucy to more sparse domains which may
  benefit from symmetry detection
• Find other applications of partition refinementa
  surprisingly general framework for distinguishing
  objects in a finite domain

19
Thank You!
20
saucy Exploiting Structure

• Graphs from typical CNF formulas possess a
  particular structure
• By construction, clauses are never connected to
  other clauses
• Thus, when refining the partition with a color of
  clauses, we can ignore all colors containing
  clauses, since we know that the connection count
  for every vertex will be zero
• Such graphs are also very sparse
• Few literals are connected to most clauses
• Few clauses are connected to most literals
• Thus, we aggressively avoid work by maintaining
  data structures (like adjacency lists) which
  explicitly direct the search and refinement
  procedures

21
saucy Exploiting Structure
1
1
1
1
1
1
1
1
1
1
1
1
22
saucy example Hole-3
23
saucy Exploiting Structure

• We can generalize this idea of avoiding obviously
  disconnected colors
• saucy improvement 3
• Iterate over a color's adjacency lists to
  determine connected colors
• Compute connection counts only for those colors

24
saucy example Hole-3

• nauty
• Iterate over all colors
• For each vertex, count connections to refining
  color, and sort
• saucy improvement 3
• Determine directly connected colors
• For each vertex, count connections to refining
  color, and sort
• saucy improvement 4
• For each vertex in refining color, count
  connections
• For every color touched, sort the counts

25
saucy Exploiting Structure

• nauty works very well on small graphs (and thus
  small formulas) but fails to scale
• Takes considerably longer than the SAT solver
  after adding SBPs
• Runs out of memory on formulas with corresponding
  graphs having gt50,000 vertices
• saucy improvement 1 sparse representation
• Input graph is represented in adjacency-list
  format

26
saucy Exploiting Structure

• Graphs from CNF formulas possess a particular
  structure
• Clause vertices only connected to their literals
• Never connected to each other
• saucy improvement 2 ignore colors containing
  clauses when refining with clauses
• We know that the connection count for every
  vertex will be zero

27
Symmetry Detection Problem

• What precisely is a symmetry of a graph G?
• A symmetry is a permutation ? of the labels
  assigned to vertices of G such that G? G
• The set of all symmetries is denoted Aut(G)

G? G
G? ? G

• (a,c',b')(a',c,b)

• (a,a')(b,c')(b',c)(C1,C2)

28
The Search Tree

• We have a stable ordered partition ? of the
  vertices of the graph. How can we extract
  Aut(G)?
• Recall the naïve approach we need labelings
  (i.e. discrete colorings)
• We select a non-singleton color T (for target)
  and generate T colorings, each with one element
  of T individualized "in front of" the remainder
  of T
• Partitions the set of all discrete colorings
  descendant from ?
• We can then recursively apply partition
  refinement to further prune the search space!
• Fix the first discrete coloring found as ? the
  remaining discrete colorings yield likely
  candidates for Aut(G)

29
The Search Tree
0
1
2
4
3
5
? 035421

• Out of 6! 720 possible labelings, partition
  refinement pruned away 714, leaving the six
  symmetries of the graph
• Discrete colorings do not necessarily yield
  symmetries
• Refinement is only an approximation
• Only occurs on highly regular graphs, which are
  uncommon in EDA applications

?
?1 (2,4)(3,5)
?3 (0,3,5)(1,2,4)
?0 ?
?5 (0,5)(1,4)
?2 (0,3)(1,2)
?4 (0,5,3)(1,4,2)
30
saucy Empirical Performance
31
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32
saucy Empirical Performance
33
saucy Empirical Performance
34
saucy Runtime Performance

• Speedup is roughly linear in the number of
  vertices
• Primarily due to efficient use of sparsity within
  the partition refinement procedure
• Search tree maintenance has relatively low
  overhead

• We ran saucy and nauty on the complement graphs
• Isomorphic search trees!
• Isolate performance difference in refinement
• Slowdown is roughly linear, which is expected
  given difference in representation

35
Ordered Partitions

• The partition refinement algorithm based on
  degree operates independently of the labeling of
  the graph
• To guarantee identical partition representations
  for isomorphic graphs, we impose an ordering on
  the colors in the partition
• When a color is split, the new colors are
  assigned in sorted order of degree with the
  refining color
• Refining colors are chosen based on position
  within the partition ordering, not based on label
• The search algorithm absolutely the refinement
  procedure to be labeling-independent!

36
Partition Refinement

• In a stable coloring
• vertices in different colors definitely cannot
  map into each other in some symmetry of the graph
• vertices in the same color may map into each
  other (i.e. refinement is only an approximation)

C1

• Refinement can't distinguish any vertices (all
  have degree two)
• Vertices in the triangle and square cannot map
  into each other
• Fortunately, this rarely happens with EDA
  instances
• An exact, polynomial time partition refinement
  algorithm would prove that the graph isomorphism
  problem is in P

c
b
a'
C3
b'
c'
a
C2
37
Additional Pruning Methods

• Enumerating all symmetries is not an option
• Aut(G) is O(n!)
• Many EDA-related instances possess exponentially
  many symmetries
• Group theory provides the answer generators
• Find a set H ? Aut(G) that implicitly represents
  the entire set of symmetries
• Every element of Aut(G) is a product
  (composition) of integer powers of elements of H
• Exponential compression H ? log2Aut(G)
• We prune away subtrees guaranteed to yield
  symmetries that we can already generate with
  previously discovered symmetries

38
Symmetry Breaking

• On all but the synthetic Urquhart instances,
  symmetry detection with nauty dominates the run
  time of the Shatter flow
• Further improvements must come from improved
  symmetry detection