Andreas Alpers, Cornell University Infeasible Systems: The Discrete - PowerPoint PPT Presentation

Andreas Alpers, Cornell University Infeasible Systems: The Discrete Tomography Polytope & The Feasible Subsystem Polytope Aussois, 01/11/06 – p.1

Andreas Alpers, Cornell University Infeasible Systems: The Discrete Tomography Polytope joint work with Peter Gritzmann, TU Munich, Germany & The Feasible Subsystem Polytope work in progress with Leslie Trotter, Cornell University Aussois, 01/11/06 – p.1

PART I The Discrete Tomography Polytope Aussois, 01/11/06 – p.2

Discrete Tomography 2 0 3 3 5 4 4 2 2 1 Aussois, 01/11/06 – p.3

Discrete Tomography 2 0 3 3 5 4 4 2 2 1 Aussois, 01/11/06 – p.3

Discrete Tomography 2 0 3 3 5 4 4 2 2 1 Aussois, 01/11/06 – p.3

Discrete Tomography 2 0 3 3 5 4 4 2 2 1 Aussois, 01/11/06 – p.3

Discrete Tomography 2 0 3 3 5 4 4 2 2 1 Aussois, 01/11/06 – p.3

The Tomography Matrix Aussois, 01/11/06 – p.4

Infeasibility: The Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } where • α, m ∈ N are fixed parameters • A m ∈ { 0 , 1 } M × N is a fixed “tomography matrix” • b m ∈ N M 0 Aussois, 01/11/06 – p.5

Infeasibility: The Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } where • α, m ∈ N are fixed parameters • A m ∈ { 0 , 1 } M × N is a fixed “tomography matrix” • b m ∈ N M 0 Given x ∗ ∈ P α ( b m ) ∩ { 0 , 1 } N there exists b ′ 0 with m ∈ N M m ) ∩ { 0 , 1 } N � = ∅ . m || 1 ≥ 2 ( m − 1 ) such that P α ( b ′ || b m − b ′ Aussois, 01/11/06 – p.5

Infeasibility: The Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } where • α, m ∈ N are fixed parameters • A m ∈ { 0 , 1 } M × N is a fixed “tomography matrix” • b m ∈ N M 0 Given x ∗ ∈ P α ( b m ) ∩ { 0 , 1 } N there exists b ′ 0 with m ∈ N M m ) ∩ { 0 , 1 } N � = ∅ . m || 1 ≥ 2 ( m − 1 ) such that P α ( b ′ || b m − b ′ Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ Aussois, 01/11/06 – p.5

Infeasibility: The Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } where • α, m ∈ N are fixed parameters • A m ∈ { 0 , 1 } M × N is a fixed “tomography matrix” • b m ∈ N M 0 Given x ∗ ∈ P α ( b m ) ∩ { 0 , 1 } N there exists b ′ 0 with m ∈ N M m ) ∩ { 0 , 1 } N � = ∅ . m || 1 ≥ 2 ( m − 1 ) such that P α ( b ′ || b m − b ′ Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ (and b ′ m � = b m ) Aussois, 01/11/06 – p.5

Infeasibility: Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m ) = ∅ ∀ b ′ m with || b m − b ′ Aussois, 01/11/06 – p.6

Infeasibility: Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m ) = ∅ ∀ b ′ m with || b m − b ′ • Even conv { x ∈ R N : A m x = b ′ 1 T x = α } = ∅ . m , 1 Aussois, 01/11/06 – p.6

Infeasibility: Discrete Tomography Polytope P α ( b m ) := conv { x ∈ { 0 , 1 } N : A m x = b m , 1 1 T x = α } Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m ) = ∅ ∀ b ′ m with || b m − b ′ • Even conv { x ∈ R N : A m x = b ′ 1 T x = α } = ∅ . m , 1 • Members of P α ( b ′ m ) with || b m − b ′ m || = 2 ( m − 1 ) give solutions to an old Number Theory problem. → (Prouhet-Tarry-Escott problem) ֒ Aussois, 01/11/06 – p.6

First Step in the Proof: LP Duality Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ ′ T u 0 T x min max → → b A T u ( LP )( b ′ ) b ′ ( DLP )( b ′ ) = = 0 Ax R N R M ∈ ∈ x u Aussois, 01/11/06 – p.7

First Step in the Proof: LP Duality Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ ′ T u 0 T x min max → → b A T u ( LP )( b ′ ) b ′ ( DLP )( b ′ ) = = 0 Ax R N R M ∈ ∈ x u • “Construct” feasible u such that b ′ T u → ∞ . Aussois, 01/11/06 – p.7

First Step in the Proof: LP Duality Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ ′ T u 0 T x min max → → b A T u ( LP )( b ′ ) b ′ ( DLP )( b ′ ) = = 0 Ax R N R M ∈ ∈ x u • “Construct” feasible u such that b ′ T u → ∞ . • “Assign values to lines s.t. they sum up to 0 (in every lattice point).” Aussois, 01/11/06 – p.7

First Step in the Proof: LP Duality Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ ′ T u 0 T x min max → → b A T u ( LP )( b ′ ) b ′ ( DLP )( b ′ ) = = 0 Ax R N R M ∈ ∈ x u • “Construct” feasible u such that b ′ T u → ∞ . • “Assign values to lines s.t. they sum up to 0 (in every lattice point).” • Since b ′ = b + ε we have b ′ T u = ε T u . Aussois, 01/11/06 – p.7

Second Step in the Proof: Polynomials Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ How to assign these values for any realization of ε ? Aussois, 01/11/06 – p.8

Second Step in the Proof: Polynomials Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ How to assign these values for any realization of ε ? ( X, ω 2 ( X )) (1,-1) (3,3) (5,15) (0,0) (2,0) (4,8) ( − Y, ω 1 ( − Y )) −1 (-5,35) +1 (-4,24) (-5,-22.5) (-3,15) (-4,-16) −1 (-2,6) (-3,-10.5) (-1,3) (-2,-6) +1 (0,0) (-1,-2.5) (0,0) (-10,-50) (-8,-32) (1,1.5) (2,2) (-6,-18) (4,0) (-4,-8) (3,1.5) (5,-2.5) (-2,-2) (0,0) ( X − Y, ω 3 ( X − Y )) ( − X − Y, ω 4 ( − X − Y )) Aussois, 01/11/06 – p.8

Last Step in the Proof: Number Theory Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ Suppose ε T u �→ ∞ , i.e., ε T u = 0 Aussois, 01/11/06 – p.9

Last Step in the Proof: Number Theory Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ Suppose ε T u �→ ∞ , i.e., ε T u = 0 − 1 · ω 1 ( − 5 ) − 1 · ω 1 ( − 2 ) + 1 · ω 1 ( − 4 ) + 1 · ω 1 ( 0 ) = 0 for any choice of ω 1 ( t ) = t k , k = 1 , . . . , m − 2 Aussois, 01/11/06 – p.9

Last Step in the Proof: Number Theory Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ Suppose ε T u �→ ∞ , i.e., ε T u = 0 − 1 · ω 1 ( − 5 ) − 1 · ω 1 ( − 2 ) + 1 · ω 1 ( − 4 ) + 1 · ω 1 ( 0 ) = 0 for any choice of ω 1 ( t ) = t k , k = 1 , . . . , m − 2 ( − 4 ) k + 0 k = ( − 5 ) k + ( − 2 ) k for k = 1 , . . . , m − 2 Aussois, 01/11/06 – p.9

Last Step in the Proof: Number Theory Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ Suppose ε T u �→ ∞ , i.e., ε T u = 0 − 1 · ω 1 ( − 5 ) − 1 · ω 1 ( − 2 ) + 1 · ω 1 ( − 4 ) + 1 · ω 1 ( 0 ) = 0 for any choice of ω 1 ( t ) = t k , k = 1 , . . . , m − 2 ( − 4 ) k + 0 k = ( − 5 ) k + ( − 2 ) k for k = 1 , . . . , m − 2 “Famous” Prouhet-Tarry-Escott problem: Fix the degree, find sets with this property. Aussois, 01/11/06 – p.9

Last Step in the Proof: Number Theory Theorem: P α ( b ′ m || 1 < 2 ( m − 1 ) . m with || b m − b ′ m ) = ∅ ∀ b ′ Suppose ε T u �→ ∞ , i.e., ε T u = 0 − 1 · ω 1 ( − 5 ) − 1 · ω 1 ( − 2 ) + 1 · ω 1 ( − 4 ) + 1 · ω 1 ( 0 ) = 0 for any choice of ω 1 ( t ) = t k , k = 1 , . . . , m − 2 ( − 4 ) k + 0 k = ( − 5 ) k + ( − 2 ) k for k = 1 , . . . , m − 2 “Famous” Prouhet-Tarry-Escott problem: Fix the degree, find sets with this property. It’s known that there are no solutions of this size. � Aussois, 01/11/06 – p.9

The Prouhet-Tarry-Escott Problem [Prouhet, 1851; Tarry, 1910; Escott, 1912]: Given k, n ∈ N . Do there exist sets { x 1 , . . . , x n } � = { y 1 , . . . , y n } ⊂ Z , s.t. x 1 1 + x 1 2 + · · · + x 1 y 1 1 + y 1 2 + · · · + y 1 = n n x 2 1 + x 2 2 + · · · + x 2 y 2 1 + y 2 2 + · · · + y 2 = n n . . . . . . x k 1 + x k 2 + · · · + x k y k 1 + y k 2 + · · · + y k = n n holds? Aussois, 01/11/06 – p.10

The Prouhet-Tarry-Escott Problem 22 25 15 27 11 -26 6 21 5 17 7 10 Aussois, 01/11/06 – p.11

The Prouhet-Tarry-Escott Problem 22 25 15 27 11 -26 6 21 5 17 7 10 { 26 , 25 , 17 , 15 , 7 , 6 } 5 = { 27 , 22 , 21 , 11 , 10 , 5 } . Aussois, 01/11/06 – p.11

PART II The Feasible Subsystem Polytope Aussois, 01/11/06 – p.12

The Feasible Subsystem Polytope M AX FS: Given an infeasible system Σ { Ax ≤ b } with A ∈ R m × n and b ∈ R m , find a feasible subsystem containing as many inequalities as possible. Aussois, 01/11/06 – p.13