Exploiting Symmetry
in Linear Programming*
Jayant Apte
ASPITRG
*Katrin Herr, R. Bödi, Symmetries in linear and integer linear programming,
Oberwolfach Report 38/2010
1
Outline -Part I
2
Outline -Part I
3
Linear Programs
4
Linear Programs
5
Linear Programs
6
Outline -Part I
7
Permutations of a set
8
Permutations of
9
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications.
Boston, MA: PWS Pub., 1994. Print.
10
The Caylay Table for symmetries
of equilateral triangle
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications.
Boston, MA: PWS Pub., 1994. Print.
11
The Cayley Table for symmetries
of equilateral triangle
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications.
Boston, MA: PWS Pub., 1994. Print.
12
Time to be more rigorous:
Groups and Group Actions
13
Group
14
Group
15
Examples of groups
16
Properties of Groups
17
Subgroups
18
Permutation Group/Symmetry
Group
19
Disjoint Cycle Notation
20
Transposition
21
Group Actions
22
G-equivalence
23
Orbits
24
Fixed point sets
25
Stabilizer Subgroup
26
Kernel of the action
27
Kernel of the action
28
Cosets
29
Cosets
30
Cosets
31
Normal Subgroups
32
Semidirect product
33
Semidirect product
34
Outline -Part I
35
Symmetries of an LP
36
Symmetries of an LP
37
What about integer programs?
38
What about integer programs?
In general, symmetries of LPs and IPs don't coincide
39
Symmetries of an integer program
40
Symmetries of an integer program
41
Symmetries of an integer program
42
Relationship between
symmetries of IP and LP
43
44
45
End of part I
●
Questions?
46
What happens to symmetries
when we add extra inequalities
to the system?
Consider the following system system of linear inequalities:
47
What happens to symmetries
when we add extra inequalities
to the system?
Consider the following system system of linear inequalities:
It is made up of 2 different systems of inequalities:
and
48
Symmetries of combined system
49
Symmetries of combined system
50
Symmetries of combined system
51
Symmetries of combined system
52
Proof
53
Proof
54
Proof
55
Proof
Row permutation we need to prove above theorem
56
Part-II
57
Part-II
58
Orbits
59
Orbits
Feasibility and Orbits
60
Orbits
Feasibility and Orbits
Why?
61
Utility and orbits
62
Utility and orbits
63
Utility and orbits
64
Orbits of bases
65
Orbits of bases
66
Structure of cost vector
67
Example
68
Part-II
69
The set of Fixed Points
70
The set of Fixed Points
71
The set of Fixed Points
72
The set of Fixed Points
Why?
73
The set of Fixed Points
74
The set of Fixed Points
75
76
77
78
79
80
Part 3
●
Prove a general relationship between number of
orbits of the set of standard basis vectors
and the dimension of subspace fixed points
●
●
●
Equivalence classes among feasible points of an
LP based on their utility value
Prove that for every feasible point there is a fixed
point with same utility value
How to formulate smaller LP given a large
symmetric LP and its symmetry group
81
Reboot
82
83
Part 3
●
Prove a general relationship between number of
orbits of the set of standard basis vectors
and the dimension of subspace fixed points
●
●
●
Equivalence classes among feasible points of an
LP based on their utility value
Prove that for every feasible point there is a fixed
point with same utility value
How to formulate smaller LP given a large
symmetric LP and its symmetry group
84
85
Whats this symbol here?
86
Direct sums
87
Direct sums
88
Direct sums
89
90
91
92
93
94
95
96
97
98
99
Part 3
●
Prove a general relationship between number of
orbits of the set of standard basis vectors
and the dimension of subspace fixed points
●
●
●
Equivalence classes among feasible points of an
LP based on their utility value
Prove that for every feasible point there is a fixed
point with same utility value
How to formulate smaller LP given a large
symmetric LP and its symmetry group
100
Example
Figure credits: Katrin Herr, R. Bödi, Symmetries in linear and integer linear programming, Oberwolfach
Report 38/2010
101
Example
102
Part 3
●
Prove a general relationship between number of
orbits of the set of standard basis vectors
and the dimension of subspace fixed points
●
●
●
Equivalence classes among feasible points of an
LP based on their utility value
Prove that for every feasible point there is a fixed
point with same utility value
How to formulate smaller LP given a large
symmetric LP and its symmetry group
103
Orbit-Stabilizer theorem
104
Orbit-Stabilizer theorem
105
Barycenter of an orbit
106
Barycenter of an orbit
107
Barycenter of an orbit
108
Barycenter of an orbit
109
Representative of equivalence class of
points having same cost in fixed space
110
Representative of equivalence class of
points having same cost in fixed space
111