Symmetry in Finite Combinatorial Objects: Scalable Methods and Applications

1. Symmetry in Finite Combinatorial Objects: Scalable Methods and Applications HadiKatebi PhD Thesis Presentation Department of Computer Science and Engineering University of Michigan April 26, 2013

2. Outline • Finite combinatorial objects • Graph Modeling • Symmetries • Importance of symmetries • Contributions • Saucy: graph symmetry-detection algorithms • Saucy+Bliss: graph canonical-labeling algorithms • Symmetries of other combinatorial objects • Boolean functions • Conclusions

3. Finite Combinatorial Objects • Objects that can be rearranged/permuted • Molecule • DNA sequence • Boolean formula • Graph • Dice • Deck of card • … F B F F

4. Graph Symmetry (Automorphism) • A permutation of the graph vertices that preserves the graph edge relation

5. Motivation and Applications • Scale combinatorial search algorithms by pruning symmetric subspaces • Constraint solving (SAT, SMT, …) • Graph canonical labeling • Build a repository of unique objects • Check equivalence through quick matching

6. Contributions • Formulated symmetry detection as search in permutation space • OPP data structure • Introduced simultaneous partition refinement that anticipates conflicts • Advantage on the graphs that previously impede leading graph-symmetry detection algorithms • Integrated our symmetry-detection framework within canonical labeling algorithms • Use symmetries to prune the search for unique signatures • Extended the symmetry-detection framework to detect functional symmetries of Boolean functions

7. Graph Symmetry-Detection and Canonical-Labeling Tools Graph Symmetry-Detection Tools nauty (1981) saucy1 (2004) saucy2 (2008) saucy3 (2012) bliss.50 (2007) bliss.72 (2011) Graph Canonical-Labeling Tools nishe (2008) saucy+bliss (2012) traces (2008)

8. Graph Symmetry Detection • Given undirected graph • with vertex set • and an initial vertex coloring • with unknown set of symmetries • Find a set of symmetries • such that generates • and

9. Group • Set with binary operation that satisfies the following group axioms: • Closure: • Associativity: • Identity: • Inverse: • Example: the set of all integers with + forms a group

10. Symmetry Group • The set of all symmetries with functional composition forms a group • Functional composition: = 1 (0 1 2

11. Group Generators • A subset of a group that generates the entire group by repeatedly composing its elements

12. Computational Complexity • Graph isomorphism: edge-preserving bijection • Automorphism is isomorphism of a graph to itself • Graph isomorphism is in NP • “Yes” answers can be verified in polynomial time on a deterministic Turing machine • Below are poly-time equivalent to graph isomorphism • Finding an isomorphism • Finding a non-trivial automorphism • Finding generators for the automorphism group

13. Computational Complexity its decision and counting versions are polynomial-time equivalent [Mathon-1979] random graphs [Babai et. al.-1980] bounded-degree graphs [Luks-1982] graphs of bounded eigenvalue multiplicity [Fürer-1995] Trees NP-complete NP Graph Isomorphism P Existing tools are fairly fast and scalable

14. Ordered Partitions • Unit partition:  • Discrete Partition: 

15. Ordered Partition Pairs (OPPs) • and are ordered partitions • : top partition • : bottom partition • Implicitly represents set of permutations

16. Discrete OPP: Single Permutation

17. Unit OPP: Permutation

18. Isomorphic OPP: Non-empty Set of Permutations

19. Matching OPP: Single Permutation Composed with a Set of Permutations Isomorphic

20. Non-isomorphic OPP: Empty Set of Permutations • or

21. Basic Search for Symmetries ü ü X X X X

22. Partition Refinement • Distinguish vertices that are not symmetric • For each vertex, compute a neighbor-count tuple • Partition vertices based on these tuples • Repeat until partition stabilizes R

23. Search Tree with Partition Refinement R ü ü

24. Isomorphic Refinement R R ü

25. Non-isomorphic Refinement R X R

26. Symmetry Group Generators • 7 vertices: 7! = 5040 permutations • 48 symmetries: 8 for square 6 for triangle • Explicit enumeration is inefficient! • Remedy: look for symmetry group generators

27. Cosets • A subgroup of a group partitions into cosets • Each coset has the same number of elements as • can be generated by composing a single representative from each coset with

28. Orbit Partition • Symmetry group induces an equivalence relation on set of vertices: the orbit partition • The orbit partition is built up incrementally from the set of found symmetries

29. Search for Group Generators x

30. Saucy 3.0 Search Tree R R R // // Orbit Partition // Coset pruning // Orbit pruning Matching OPP Non-isomorphic OPP

31. Simultaneous Partition Refinement A Part of Miyazaki Graph Conventional Simultaneous 4 gens 4 gens 1gen 1gen

32. Refinement Before 144 Color 14 differently & propagate Color 4 differently & propagate

33. Conflict Anticipation Color 14 differently & propagate Color 4 differently & propagate

34. Overlooked Conflict This OPP violates the edge relation and is a conflict!

35. Simultaneous Refinement Color 14 differently & propagate Color 4 differently & propagate

36. Search Tree with Conventional Refinement (saucy 2.0) 4 gens 1gen Orbit pruned 16 conflicts

37. Search Tree with Simultaneous Refinement (saucy 3.0) 4 gens Orbit pruned 1gen 4 conflicts

38. Saucy 3.0 vs. Saucy 2.0 Runtime:Miyazaki Graphs

39. Saucy 3.0 vs. Bliss 0.72

40. Graph Canonical Labeling • A signature is the lexicographically minimal listing of the graph edges • Graph symmetries preserve signatures • #labelings = 4! = 24 • #symmetries = 8 • #distinct signatures = 24/8 = 3 0 3 2 1 2 1 0 3 1 0 3 2 1 0 3 2 3 2 1 0 3 2 1 0 2 1 0 3 0 3 2 1 0 2 3 1 0 1 3 2 1 0 2 3 2 0 1 3 2 3 1 0 1 3 2 0 3 1 0 2 3 2 0 1 0 3 1 2 0 2 1 3 2 0 3 1 3 0 2 1 3 1 2 0 2 1 3 0 1 2 0 3 1 3 0 2

41. Bliss Search for a Canonical Labeling 1 2 1 2 1 2 1 2 1 2 1 2

42. Structure of Canonical-Labeling Tree

43. Two-pass Canonical Labeling bliss 0.72 saucy 3.0 Sequence of Stabilizers Symmetry Search Routines Sequence of Individualizations Input Graph Canonical Labeling Search Routines Orbit Partitions at Each Coset Level Canonical Labeling

44. Canonical Labeling Runtimes

45. Symmetries of Boolean Functions • Extend saucy framework to search for the symmetries of Boolean functions • Capture partial functionality by abstraction graphs • Dependency graph • Simulation graphs • Use abstraction graphs to prune the search space • Use SAT to check equivalence under candidate permutations • Learn from SAT counterexamples to avoid similar conflicts

46. Conclusions • Saucy 3.0: most scalable graph symmetry detection software package • Simultaneous partition refinement • Saucy 3.0 + Bliss 0.72: most scalable graph canonical labeling software package • Used Saucy as a preprocessing step for Bliss • Extend the Saucy framework to search for symmetries of other combinatorial objects • Ex., Boolean functions

47. Publications • HadiKatebi, Karem A. Sakallah, Igor L. Markov: “Generalized Boolean Symmetries Through Nested Partition Refinement." submitted to ICCAD 2013. • HadiKatebi, Karem A. Sakallah and Igor L. Markov: "Graph Symmetry Detection and Canonical Labeling: Differences and Synergies." in Proc. Turing-100 2012. (Best Paper Award) • HadiKatebi, Karem A. Sakallah and Igor L. Markov: "Conflict Anticipation in the Search for Graph Automorphisms." in Proc. LPAR 2012. • HadiKatebi, Karem A. Sakallah, Igor L. Markov: "Symmetry and Satisfiability: An Update." in Proc. SAT 2010. • HadiKatebi and Igor L. Markov: "Large-scale Boolean Matching." In Advanced Techniques in Logic Synthesis, Optimizations and Applications, S. Khatri and K. Gulati, eds; Springer, 2011. • HadiKatebi, Igor L. Markov: "Large-scale Boolean matching." in Proc. Of DATE 2010

48. Thanks for listening Please ask questions 