Post on 15-Jul-2020
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Exploiting Symmetry in Linear Programming*
Jayant ApteASPITRG
*Katrin Herr, R. Bödi, Symmetries in linear and integer linear programming, Oberwolfach Report 38/2010
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Outline-Part I
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Outline-Part I
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Linear Programs
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Linear Programs
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Linear Programs
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Outline-Part I
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Permutations of a set
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Permutations of
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Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications. Boston, MA: PWS Pub., 1994. Print.
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The Caylay Table for symmetries of equilateral triangle
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications. Boston, MA: PWS Pub., 1994. Print.
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The Cayley Table for symmetries of equilateral triangle
Figure Credits: Judson, Thomas W. Abstract Algebra: Theory and Applications. Boston, MA: PWS Pub., 1994. Print.
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Time to be more rigorous:Groups and Group Actions
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Group
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Group
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Examples of groups
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Properties of Groups
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Subgroups
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Permutation Group/Symmetry Group
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Disjoint Cycle Notation
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Transposition
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Group Actions
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G-equivalence
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Orbits
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Fixed point sets
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Stabilizer Subgroup
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Kernel of the action
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Kernel of the action
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Cosets
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Cosets
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Cosets
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Normal Subgroups
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Semidirect product
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Semidirect product
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Outline-Part I
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Symmetries of an LP
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Symmetries of an LP
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What about integer programs?
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What about integer programs?
In general, symmetries of LPs and IPs don't coincide
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Symmetries of an integer program
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Symmetries of an integer program
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Symmetries of an integer program
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Relationship between symmetries of IP and LP
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End of part I
● Questions?
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What happens to symmetries when we add extra inequalities
to the system?
Consider the following system system of linear inequalities:
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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
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Symmetries of combined system
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Symmetries of combined system
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Symmetries of combined system
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Symmetries of combined system
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Proof
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Proof
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Proof
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Proof
Row permutation we need to prove above theorem
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Part-II
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Part-II
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Orbits
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Orbits
Feasibility and Orbits
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Orbits
Feasibility and Orbits
Why?
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Utility and orbits
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Utility and orbits
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Utility and orbits
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Orbits of bases
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Orbits of bases
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Structure of cost vector
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Example
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Part-II
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The set of Fixed Points
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The set of Fixed Points
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The set of Fixed Points
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The set of Fixed Points
Why?
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The set of Fixed Points
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The set of Fixed Points
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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
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Reboot
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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
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Whats this symbol here?
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Direct sums
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Direct sums
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Direct sums
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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
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Example
Figure credits: Katrin Herr, R. Bödi, Symmetries in linear and integer linear programming, Oberwolfach Report 38/2010
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Example
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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
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Orbit-Stabilizer theorem
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Orbit-Stabilizer theorem
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Barycenter of an orbit
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Barycenter of an orbit
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Barycenter of an orbit
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Barycenter of an orbit
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Representative of equivalence class of points having same cost in fixed space
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Representative of equivalence class of points having same cost in fixed space