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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

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SLIDE 1

Andreas Alpers, Cornell University Infeasible Systems: The Discrete Tomography Polytope & The Feasible Subsystem Polytope

Aussois, 01/11/06 – p.1

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SLIDE 2

Andreas Alpers, Cornell University Infeasible Systems: The Discrete Tomography Polytope & The Feasible Subsystem Polytope

joint work with Peter Gritzmann, TU Munich, Germany work in progress with Leslie Trotter, Cornell University

Aussois, 01/11/06 – p.1

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SLIDE 3

PART I

The Discrete Tomography Polytope

Aussois, 01/11/06 – p.2

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SLIDE 4

Discrete Tomography

2 3 3 5 1 2 2 4 4

Aussois, 01/11/06 – p.3

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SLIDE 5

Discrete Tomography

2 3 3 5 1 2 2 4 4

Aussois, 01/11/06 – p.3

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SLIDE 6

Discrete Tomography

2 4 4 2 3 3 5 1 2

Aussois, 01/11/06 – p.3

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SLIDE 7

Discrete Tomography

4 2 3 3 5 1 2 2 4

Aussois, 01/11/06 – p.3

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SLIDE 8

Discrete Tomography

4 2 3 3 5 1 2 2 4

Aussois, 01/11/06 – p.3

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SLIDE 9

The Tomography Matrix

Aussois, 01/11/06 – p.4

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SLIDE 10

Infeasibility: The Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

where

  • α, m ∈ N are fixed parameters
  • Am ∈ {0, 1}M×N is a fixed “tomography matrix”
  • bm ∈ NM

Aussois, 01/11/06 – p.5

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SLIDE 11

Infeasibility: The Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

where

  • α, m ∈ N are fixed parameters
  • Am ∈ {0, 1}M×N is a fixed “tomography matrix”
  • bm ∈ NM

Given x∗ ∈ Pα(bm) ∩ {0, 1}N there exists b′

m ∈ NM 0 with

||bm − b′

m||1 ≥ 2(m − 1) such that Pα(b′ m) ∩ {0, 1}N = ∅.

Aussois, 01/11/06 – p.5

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SLIDE 12

Infeasibility: The Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

where

  • α, m ∈ N are fixed parameters
  • Am ∈ {0, 1}M×N is a fixed “tomography matrix”
  • bm ∈ NM

Given x∗ ∈ Pα(bm) ∩ {0, 1}N there exists b′

m ∈ NM 0 with

||bm − b′

m||1 ≥ 2(m − 1) such that Pα(b′ m) ∩ {0, 1}N = ∅.

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Aussois, 01/11/06 – p.5

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SLIDE 13

Infeasibility: The Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

where

  • α, m ∈ N are fixed parameters
  • Am ∈ {0, 1}M×N is a fixed “tomography matrix”
  • bm ∈ NM

Given x∗ ∈ Pα(bm) ∩ {0, 1}N there exists b′

m ∈ NM 0 with

||bm − b′

m||1 ≥ 2(m − 1) such that Pα(b′ m) ∩ {0, 1}N = ∅.

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

(and b′

m = bm)

Aussois, 01/11/06 – p.5

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SLIDE 14

Infeasibility: Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Aussois, 01/11/06 – p.6

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SLIDE 15

Infeasibility: Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

  • Even conv{x ∈ RN : Amx = b′

m, 1

1Tx = α} = ∅.

Aussois, 01/11/06 – p.6

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SLIDE 16

Infeasibility: Discrete Tomography Polytope

Pα(bm) := conv{x ∈ {0, 1}N : Amx = bm, 1

1Tx = α}

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

  • Even conv{x ∈ RN : Amx = b′

m, 1

1Tx = α} = ∅.

  • Members of Pα(b′

m) with ||bm − b′ m|| = 2(m − 1)

give solutions to an old Number Theory problem.

֒ → (Prouhet-Tarry-Escott problem)

Aussois, 01/11/06 – p.6

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SLIDE 17

First Step in the Proof: LP Duality

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

0Tx →

min

b

′Tu

→

max

(LP)(b′) Ax = b′ (DLP)(b′) ATu = x ∈ RN u ∈ RM

Aussois, 01/11/06 – p.7

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SLIDE 18

First Step in the Proof: LP Duality

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

0Tx →

min

b

′Tu

→

max

(LP)(b′) Ax = b′ (DLP)(b′) ATu = x ∈ RN u ∈ RM

  • “Construct” feasible u such that b

′Tu → ∞.

Aussois, 01/11/06 – p.7

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SLIDE 19

First Step in the Proof: LP Duality

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

0Tx →

min

b

′Tu

→

max

(LP)(b′) Ax = b′ (DLP)(b′) ATu = x ∈ RN u ∈ RM

  • “Construct” feasible u such that b

′Tu → ∞.

  • “Assign values to lines s.t. they sum up to 0

(in every lattice point).”

Aussois, 01/11/06 – p.7

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SLIDE 20

First Step in the Proof: LP Duality

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

0Tx →

min

b

′Tu

→

max

(LP)(b′) Ax = b′ (DLP)(b′) ATu = x ∈ RN u ∈ RM

  • “Construct” feasible u such that b

′Tu → ∞.

  • “Assign values to lines s.t. they sum up to 0

(in every lattice point).”

  • Since b′ = b + ε we have b

′Tu = εTu.

Aussois, 01/11/06 – p.7

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SLIDE 21

Second Step in the Proof: Polynomials

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

How to assign these values for any realization of ε?

Aussois, 01/11/06 – p.8

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SLIDE 22

Second Step in the Proof: Polynomials

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

How to assign these values for any realization of ε?

−1 −1 +1 +1

(5,-2.5) (0,0) (2,0) (4,8) (5,15) (3,3) (1,-1) (-5,-22.5) (-4,-16) (-3,-10.5) (-2,-6) (-1,-2.5) (0,0) (1,1.5) (0,0) (-2,-2) (-4,-8) (-6,-18) (-8,-32) (-10,-50)

(X, ω2(X)) (−Y, ω1(−Y ))

(-5,35) (-4,24) (-3,15) (-2,6) (-1,3) (0,0)

(−X − Y, ω4(−X − Y )) (X − Y, ω3(X − Y ))

(2,2) (4,0) (3,1.5)

Aussois, 01/11/06 – p.8

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SLIDE 23

Last Step in the Proof: Number Theory

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Suppose εTu → ∞, i.e., εTu = 0

Aussois, 01/11/06 – p.9

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SLIDE 24

Last Step in the Proof: Number Theory

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Suppose εTu → ∞, i.e., εTu = 0

−1 · ω1(−5) − 1 · ω1(−2) + 1 · ω1(−4) + 1 · ω1(0) = 0

for any choice of ω1(t) = tk, k = 1, . . . , m − 2

Aussois, 01/11/06 – p.9

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SLIDE 25

Last Step in the Proof: Number Theory

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Suppose εTu → ∞, i.e., εTu = 0

−1 · ω1(−5) − 1 · ω1(−2) + 1 · ω1(−4) + 1 · ω1(0) = 0

for any choice of ω1(t) = tk, k = 1, . . . , m − 2

(−4)k + 0k = (−5)k + (−2)k for k = 1, . . . , m − 2

Aussois, 01/11/06 – p.9

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SLIDE 26

Last Step in the Proof: Number Theory

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Suppose εTu → ∞, i.e., εTu = 0

−1 · ω1(−5) − 1 · ω1(−2) + 1 · ω1(−4) + 1 · ω1(0) = 0

for any choice of ω1(t) = tk, k = 1, . . . , m − 2

(−4)k + 0k = (−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

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SLIDE 27

Last Step in the Proof: Number Theory

Theorem: Pα(b′

m) = ∅ ∀b′ m with ||bm − b′ m||1 < 2(m − 1).

Suppose εTu → ∞, i.e., εTu = 0

−1 · ω1(−5) − 1 · ω1(−2) + 1 · ω1(−4) + 1 · ω1(0) = 0

for any choice of ω1(t) = tk, k = 1, . . . , m − 2

(−4)k + 0k = (−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

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The Prouhet-Tarry-Escott Problem

[Prouhet, 1851; Tarry, 1910; Escott, 1912]:

Given k, n ∈ N. Do there exist sets {x1, . . . , xn} = {y1, . . . , yn} ⊂ Z, s.t.

x1

1 + x1 2 + · · · + x1 n

= y1

1 + y1 2 + · · · + y1 n

x2

1 + x2 2 + · · · + x2 n

= y2

1 + y2 2 + · · · + y2 n

. . . . . .

xk

1 + xk 2 + · · · + xk n

= yk

1 + yk 2 + · · · + yk n

holds?

Aussois, 01/11/06 – p.10

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SLIDE 29

The Prouhet-Tarry-Escott Problem

  • 26

15 11 6 5 7 10 17 21 27 25 22

Aussois, 01/11/06 – p.11

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SLIDE 30

The Prouhet-Tarry-Escott Problem

  • 26

15 11 6 5 7 10 17 21 27 25 22

{26, 25, 17, 15, 7, 6} 5 = {27, 22, 21, 11, 10, 5}.

Aussois, 01/11/06 – p.11

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SLIDE 31

PART II

The Feasible Subsystem Polytope

Aussois, 01/11/06 – p.12

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SLIDE 32

The Feasible Subsystem Polytope

MAX FS: Given an infeasible system Σ {Ax ≤ b} with

A ∈ Rm×n and b ∈ Rm, find a feasible subsystem

containing as many inequalities as possible.

Aussois, 01/11/06 – p.13

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SLIDE 33

The Feasible Subsystem Polytope

MAX FS: Given an infeasible system Σ {Ax ≤ b} with

A ∈ Rm×n and b ∈ Rm, find a feasible subsystem

containing as many inequalities as possible. Feasible Subsystem Polytope:

PFS := conv ({y ∈ {0, 1}m : y is the incidence vector of a

feasible subsystem of Σ})

Aussois, 01/11/06 – p.13

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SLIDE 34

The Feasible Subsystem Polytope

MAX FS: Given an infeasible system Σ {Ax ≤ b} with

A ∈ Rm×n and b ∈ Rm, find a feasible subsystem

containing as many inequalities as possible. Feasible Subsystem Polytope:

PFS := conv ({y ∈ {0, 1}m : y is the incidence vector of a

feasible subsystem of Σ})

֒ → Independence System Polytope

Aussois, 01/11/06 – p.13

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SLIDE 35

The Feasible Subsystem Polytope

MAX FS: Given an infeasible system Σ {Ax ≤ b} with

A ∈ Rm×n and b ∈ Rm, find a feasible subsystem

containing as many inequalities as possible. Feasible Subsystem Polytope:

PFS := conv ({y ∈ {0, 1}m : y is the incidence vector of a

feasible subsystem of Σ})

֒ → Independence System Polytope

[1] T.S. MOTZKIN, BEITRÄGE ZUR THEORIE DER LINEAREN UNGLEICHUNGEN, PHD THESIS, 1933 [2] M. PARKER, A SET COVERING APPROACH TO INFEASIBILITY ANALYSIS OF

LINEAR PROGRAMMING PROBLEMS AND RELATED ISSUES, PHD THESIS, 1995

[3] E. AMALDI, M.E. PFETSCH AND L.E. TROTTER, ON THE MAXIMUM FEASIBLE

SUBSYSTEM PROBLEM, IISS AND IIS-HYPERGRAPHS, MATH. PROGRAM., 2003

Aussois, 01/11/06 – p.13

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SLIDE 36

IIS

Irreducible Infeasible Subsystem Σ′: Every proper subsystem of Σ′ is feasible.

2 1 3 2 1

K K3 K

Aussois, 01/11/06 – p.14

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SLIDE 37

Known (Rank) Facets for PFS

  • s∈S ys ≤ rank (S)

Aussois, 01/11/06 – p.15

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SLIDE 38

Known (Rank) Facets for PFS

  • s∈S ys ≤ rank (S)

(1) Facet: If S is an IIS (then rank (S) = |S| − 1)

Aussois, 01/11/06 – p.15

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SLIDE 39

Known (Rank) Facets for PFS

  • s∈S ys ≤ rank (S)

(1) Facet: If S is an IIS (then rank (S) = |S| − 1) (2) Facet: If S induces an AW(|S|, |S| − 2)-antiweb (then rank (S) = |S| − 2)

Aussois, 01/11/06 – p.15

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SLIDE 40

Known (Rank) Facets for PFS

  • s∈S ys ≤ rank (S)

(1) Facet: If S is an IIS (then rank (S) = |S| − 1) (2) Facet: If S induces an AW(|S|, |S| − 2)-antiweb (then rank (S) = |S| − 2) Remarks:

Aussois, 01/11/06 – p.15

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SLIDE 41

Known (Rank) Facets for PFS

  • s∈S ys ≤ rank (S)

(1) Facet: If S is an IIS (then rank (S) = |S| − 1) (2) Facet: If S induces an AW(|S|, |S| − 2)-antiweb (then rank (S) = |S| − 2) Remarks:

  • [Laurent, ’89] Generalized antiwebs

(necessary & sufficient conditions)

Aussois, 01/11/06 – p.15

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SLIDE 42

Known (Rank) Facets for PFS

  • s∈S ys ≤ rank (S)

(1) Facet: If S is an IIS (then rank (S) = |S| − 1) (2) Facet: If S induces an AW(|S|, |S| − 2)-antiweb (then rank (S) = |S| − 2) Remarks:

  • AW(|S|, |S| − 2) only possible type of a

generalized antiweb

Aussois, 01/11/06 – p.15

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SLIDE 43

An Antiweb AW(5, 3)

S = {1, 2, 3, 4, 5} 2 1 3 4 5

Aussois, 01/11/06 – p.16

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SLIDE 44

An Antiweb AW(5, 3)

S = {1, 2, 3, 4, 5} 2 1 3 4 5

Aussois, 01/11/06 – p.16

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SLIDE 45

An Antiweb AW(5, 3)

S = {1, 2, 3, 4, 5} 2 1 3 4 5

Aussois, 01/11/06 – p.16

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SLIDE 46

An Antiweb AW(5, 3)

S = {1, 2, 3, 4, 5} 2 1 3 4 5

Aussois, 01/11/06 – p.16

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SLIDE 47

An Antiweb AW(5, 3)

S = {1, 2, 3, 4, 5} 2 1 3 4 5

Aussois, 01/11/06 – p.16

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SLIDE 48

An Antiweb AW(5, 3)

S = {1, 2, 3, 4, 5} 2 1 3 4 5

Aussois, 01/11/06 – p.16

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SLIDE 49

New Results

New class of facets: AW−(|S|, |S| − 2) “Antiweb minus 1 hyperedge”

5 1 2 3 4 2 1 3 4 5

AW− AW

Aussois, 01/11/06 – p.17

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SLIDE 50

New Results

Theorem: Given a general infeasible system

Σ : {Ax ≤ b} with A ∈ Rm×n and b ∈ Rm, and let S ⊆ {1, . . . , m}. For β ∈ N let (∗)

  • e∈S

ye ≤ β,

denote a valid inequality for PFS. According to the choice of β we have the following classification: (i) If β = 1: (∗) defines a facet if and only if |S| = 1;

Aussois, 01/11/06 – p.18

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SLIDE 51

New Results

Theorem: Given a general infeasible system

Σ : {Ax ≤ b} with A ∈ Rm×n and b ∈ Rm, and let S ⊆ {1, . . . , m}. For β ∈ N let (∗)

  • e∈S

ye ≤ β,

denote a valid inequality for PFS. According to the choice of β we have the following classification: (i) If β = 1: (∗) defines a facet if and only if |S| = 1; (ii) If 1 < β < n: (∗) never defines a facet;

Aussois, 01/11/06 – p.18

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SLIDE 52

New Results

Theorem: Given a general infeasible system

Σ : {Ax ≤ b} with A ∈ Rm×n and b ∈ Rm, and let S ⊆ {1, . . . , m}. For β ∈ N let (∗)

  • e∈S

ye ≤ β,

denote a valid inequality for PFS. According to the choice of β we have the following classification: (i) If β = 1: (∗) defines a facet if and only if |S| = 1; (ii) If 1 < β < n: (∗) never defines a facet; (iii) If β = n: (∗) defines a facet if and only if S is an IIS;

Aussois, 01/11/06 – p.18

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SLIDE 53

New Results

Theorem: Given a general infeasible system

Σ : {Ax ≤ b} with A ∈ Rm×n and b ∈ Rm, and let S ⊆ {1, . . . , m}. For β ∈ N let (∗)

  • e∈S

ye ≤ β,

denote a valid inequality for PFS. According to the choice of β we have the following classification: (i) If β = 1: (∗) defines a facet if and only if |S| = 1; (ii) If 1 < β < n: (∗) never defines a facet; (iii) If β = n: (∗) defines a facet if and only if S is an IIS; (iv) If β = n + 1: (∗) defines a facet if and only if

CS = AW(n + 3, n + 1) or CS = AW−(n + 3, n + 1).

Aussois, 01/11/06 – p.18

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SLIDE 54

Thank you for your attention!

Aussois, 01/11/06 – p.19