Adjacency decomposition method breaks up a large polyhedral representation conversion problem into several smaller representation conversion problems. Given a group G acting on the set of rays, the smaller problems are that of finding G-in-equivalent neighbors of a given extreme ray.

1. Polyhedral Description Conversion
up to Symmetries*
Jayant Apte
ASPITRG
[1] *D. Bremner, M. D. Sikiric, and A. Schurmann. Polyhedral representation conversion up to symmetries. CoRR, abs/math/0702239, 2007
[2] Thomas Rehn. Polyhedral Description Conversion up to Symmetries. Diploma thesis (mathematics), Otto von Guericke University
Magdeburg, November 2010
[3] Abstract Algebra: Theory and Applications by Thomas Judson. Available online.

2. Outline:Part-I
● Groups
● Properties of Groups
● Permutations/Symmetry Group
● Cosets and Lagrange's Theorem
● Isomorphisms and Caylay's Theorem
● Group Actions and Orbits
● Fixed point sets and stabilizers
● Face lattices of polyhedra
● Combinatorial automorphism group of polyhedra

3. Outline:Part-II
● Representation conversion problems
● Adjacency decomposition method
● Neighbors of extreme rays
● Support cone
● Reduced support cone
● Enumeration of G-inequivalent extreme rays of
reduced support cone
● Example

4. A simple example

5. Rigid Motions of
an equilateral
triangle
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications.
Boston, MA: PWS Pub., 1994. Print.

6. Rigid Motions and Symmetry

7. The Caylay Table for symmetries
of equilateral triangle
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications.
Boston, MA: PWS Pub., 1994. Print.

8. Group

9. Group

10. Examples of groups

11. Examples of groups

12. Properties of Groups

13. Properties of Groups

14. Subgroups

15. Examples of subgroups

16. Cyclic Subgroups and
Cyclic Groups

17. Examples

18. Properties of Cyclic Groups

19. Permutation Group/Symmetry
Group

20. Disjoint Cycle Notation

21. Transposition

22. Dihedral Groups

23. Example:

24. Cosets

25. Cosets

26. Cosets

27. Double Cosets

28. Double Cosets

29. Isomorphisms

30. Automorphisms

31. Group Actions

32. G-equivalence

33. Orbits

34. Fixed point sets

35. Stabilizer Subgroup

36. Groups acting on polyhedra

37. Example

38. After Homogenization:

39. d=3
d=2
d=1
d=0
Hasse Diagrams
of and
A B C D E F
AB CD DE EF AFBC
ABCDEF
Hasse Diagram of
A*Z B*Z C*Z D*Z E*Z F*Z
A*B*Z C*D*Z D*E*Z E*F*Z A*F*ZB*C*Z
A*B*C*D*E*F*Z
Z
Hasse Diagram of

40. Combinatorial Automorphism
Group of

41. Combinatorial Automorphism
Group of

42. A subgroup of combinatorial
automorphism group

43. A closer look at

44. A closer look at

45. A closer look at

46. A closer look at

47. Caylay table for G

48. Caylay table for G
G is abelian!

49. The Representation
Conversion Problem
● WLOG, polyhedra can be expressed in two
equivalent ways:
– (1) The halfspace/inequality representation (H-rep)
– (2) The extreme ray representation (V-rep)
● (Prob 1) (1)---->(2): Extreme ray enumeration
● (Prob 2) (2)---->(1): Facet enumeration
●
● Hence we can try solving (Prob 2) only
● Additionally, WLOG we can assume that input
polyhedra are actually polyhedral cones

50. Known exact algorithms
● (PM) Pivoting Methods: Use simplex method as tool Roughly speaking,
traverse a directed graph with LP bases as vertices and for every pair of
vertices, the existence of reverse simplex pivot creating a directed
edge. Recover extreme rays of the polyhedral cone via an onto mapping
from set of bases to the set of rays.
● (IM) Incremental Methods: Builds the set of facets of polyhedra formed
by successively larger set of input extreme rays
● (DM) Decomposition Methods: Decompose (Prob 2) into set of smaller
(in dimension) (Prob1)s or (Prob2)s and solve them using (PM) or (IM)

51. The adjacency decomposition
method
● Defines a notion of adjacency of rays
(neighborhood)
● Maintains a set of G-inequivalent rays of input
cone
● For every extreme ray in this set, the problem
of finding its neighbors is posed as (Prob 2)
● Symmetry is exploited by keeping track of only
the G-inequivalent neighbors

52. Neighborhood of rays

53. Neighborhood of rays

54. Support Cone

55. Reduced Support Cone

56. Extreme rays of

57. Proof contd...

58. Moral of the story

59. Computing the neighbors

60. Computing the neighbors

61. Algorithm for computing neighbors

62. Adjacency Decomposition Method

63. Questions