Outline

Edmonds-Giles theorem

K. Subramani^{1}

1Lane Department of Computer Science and Electrical Engineering West Virginia University

April 1, 2013

Subramani  Edmonds-Giles theorem

Outline

Outline

1 Fundamentals
2 The Theorem
3 Consequences to TDI systems

1 Fundamentals
2 The Theorem
3 Consequences to TDI systems

1 Fundamentals
2 The Theorem
3 Consequences to TDI systems

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Fundamentals

Definition

A polyhedron in ℝ^{n} is a set of the form: P = {x ∈ ℜ^{n} : A · x ≤ b}, for some matrix A ∈ ℜ^{m×n} and vector b ∈ ℜ^{m}. If A and b are rational, the polyhedron is said to be rational. A bounded polyhedron is called a polytope.

Definition

The dimension of any non-empty set X ⊆ ℜ^{n} i.e., dimX, is defined as:

n − max{rank(A) : A is ann × n matrix such thatA · x = A · y, ∀x, y ∈ X}.

Definition

Let P = {x : A · x ≤ b} denote a non-empty polyhedron. Let c denote a non-zero vector, such that δ = max{c · x : x ∈ P} is finite. , Then {x : c · x = δ} is called a supporting hyperplane of P. A face of P is P itself or the intersection of P with a supporting hyperplane of P. A point x for which {x} is a face, is called a vertex of P. Every vertex of P is in fact a basic feasible solution of the system A · x ≤ b. A minimal face of P is a face that does not contain any other face.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

Unimodular Transformations

Definition

Let P denote a polyhedron {x : A · x ≤ b}. We define the set P_{I} = {x : A · x ≤ b}_{I} to be the convex hull of the integral vectors in P. P_{I} is also called the integer hull of P. Clearly, P_{I} ⊆ P. A polyhedron is said to be integral, if P = P_{I}.

Definition

Let A be an arbitrary matrix. A series of the following operations:

(a) Multiplying a column by −1,

(b) Subtracting one column from another column, and

(c) Exchanging two columns

is called a unimodular transformation.

Lemma

Let A be a rational matrix and let b be a rational column vector. Then, A · x = b has an integral solution if and only if y · b is an integer, for each each rational vector y, for which y · A is integral.

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

The Edmonds-Giles Theorem

Theorem Let P be a rational polyhedron. Then the following statements are equivalent:

(a) P is integral.

(b) Each face of P contains integral vectors.

(c) Each minimal face of P contains integral vectors.

(d) Each supporting hyperplane of P contains integral vectors.

(e) Each rational supporting hyperplane of P contains integral vectors.

(f) max{c · x : x ∈ P} is attained by an integral vector y, for each c for which the maximum is finite.

(g) max{c · x : x ∈ P} is an integer, for each integral c, for which the maximum is finite.

Proof. First prove (a) ⇒ (b) ⇒ (f ) ⇒ (a). Then prove (b) ⇒ (d) ⇒ (e) ⇒ (c) ⇒ (b). Finally, prove (f ) ⇒ (g) ⇒ (e).

Fundamentals The Theorem

Consequences to TDI systems

Totally Dual Integral Systems

Definition

A polyhedral system A · x ≤ b is said to be Totally Dual Integral (TDI), if the minimum in the Linear Programming duality equation

max{c · x : A · x ≤ b} = min{y · b : y · A = c, y ≥ 0}

has an integral solution y for each integral c, for which the minimum is finite.

Theorem

TDI-theorem Let A · x ≤ b denote a TDI system, where A is rational and b is integral. Then the polyhedron {x : A · x ≤ b} is integral.

Proof.

Direct consequence of (g) ⇒ (a)!

Totally Dual Integral Systems

Definition

A polyhedral system A · x ≤ b is said to be Totally Dual Integral (TDI), if the minimum in the Linear Programming duality equation

max{c · x : A · x ≤ b} = min{y · b : y · A = c, y ≥ 0}

has an integral solution y for each integral c, for which the minimum is finite.

Theorem

TDI-theorem Let A · x ≤ b denote a TDI system, where A is rational and b is integral. Then the polyhedron {x : A · x ≤ b} is integral.

Proof.

Direct consequence of (g) ⇒ (a)!

Totally Dual Integral Systems

Definition

A polyhedral system A · x ≤ b is said to be Totally Dual Integral (TDI), if the minimum in the Linear Programming duality equation

max{c · x : A · x ≤ b} = min{y · b : y · A = c, y ≥ 0}

has an integral solution y for each integral c, for which the minimum is finite.

Theorem

TDI-theorem Let A · x ≤ b denote a TDI system, where A is rational and b is integral. Then the polyhedron {x : A · x ≤ b} is integral.

Proof.

Direct consequence of (g) ⇒ (a)!

Totally Dual Integral Systems

Definition

A polyhedral system A · x ≤ b is said to be Totally Dual Integral (TDI), if the minimum in the Linear Programming duality equation

max{c · x : A · x ≤ b} = min{y · b : y · A = c, y ≥ 0}

has an integral solution y for each integral c, for which the minimum is finite.

Theorem

TDI-theorem Let A · x ≤ b denote a TDI system, where A is rational and b is integral. Then the polyhedron {x : A · x ≤ b} is integral.

Proof.

Direct consequence of (g) ⇒ (a)!

Totally Dual Integral Systems

Definition

A polyhedral system A · x ≤ b is said to be Totally Dual Integral (TDI), if the minimum in the Linear Programming duality equation

max{c · x : A · x ≤ b} = min{y · b : y · A = c, y ≥ 0}

has an integral solution y for each integral c, for which the minimum is finite.

Theorem

TDI-theorem Let A · x ≤ b denote a TDI system, where A is rational and b is integral. Then the polyhedron {x : A · x ≤ b} is integral.

Proof.

Direct consequence of (g) ⇒ (a)!

Hoffman-Kruskal Theorem

Theorem

An integral matrix A is totally unimodular (TU), if and only if the polyhedron {x : A · x ≤ b} is integral for each integral vector b.

Hoffman-Kruskal Theorem

Theorem

An integral matrix A is totally unimodular (TU), if and only if the polyhedron {x : A · x ≤ b} is integral for each integral vector b.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.

Fundamentals The Theorem

Consequences to TDI systems

Homework II, Problem 5

Theorem

An integral matrix A is totally unimodular if and only if the system {A · x ≤ b, x ≥ 0} is TDI for each vector b.

Proof.

Only If: Assume that A is TU. It follows that A^{T} is TU. Hence, min{y · b : y · A ≥ c, y ≥ 0} is attained by an integral vector, for each vector b and each integral vector c, for which the minimum is finite (Hoffman-Kruskal)! In other words, the system {A · x ≤ b, x ≥ 0} is TDI, for each vector b.

If: Assume that the system {A · x ≤ b, x ≥ 0} is TDI for each vector b. It follows that the system is TDI for each integral vector b. By the TDI-theorem, the polyhedron {x : A · x ≤ b, x ≥ 0} is integral for each integral vector b. By the Hoffman-Kruskal theorem, it follows that A is TU.