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The Structure of Polyhedra Gabriel Indik March 2006 CAS 746 – Advanced Topics in Combinatorial Optimization

The Structure of Polyhedra

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CAS 746 – Advanced Topics in Combinatorial Optimization. The Structure of Polyhedra. Gabriel Indik March 2006. Implicit equalities Redundant constrains Characteristic/recession cone Lineality space Affine hull and dimension Supporting hyperplanes Faces Maximal faces: facets - PowerPoint PPT Presentation

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Page 1: The Structure of Polyhedra

The Structure of Polyhedra

Gabriel Indik

March 2006

CAS 746 – Advanced Topics in Combinatorial Optimization

Page 2: The Structure of Polyhedra

Implicit equalities Redundant constrains Characteristic/recession cone Lineality space Affine hull and dimension Supporting hyperplanes Faces Maximal faces: facets Minimal faces: vertices The face lattice Convex hull - extreme points (vertices) Extreme rays Decomposition of polyhedra Application

Presentation outline

Page 3: The Structure of Polyhedra

An inequality ax from Ax b is called an implicit equality (in Ax b) if ax = for all x satisfying Ax b. Notation: A= x b= is the system of implicit equalities in Ax b A+ x b+ is the system of all other inequalities in Ax b

Implicit equalities

Example:Take the following system of inequalities a, b, c and d in R2:

c

b

a

d

Inequalities b and d are tight (satisfied with equality) for all x satisfying a, b, c and d.Thus, inequalities b and d are implicit equalities in this particular system.

A polyhedron is fully dimensional there are no implicit equalities

Page 4: The Structure of Polyhedra

A constraint in a constraint system is called redundant (in the system) if it is implied by the other constraints in the system. A redundant constraint can be removed without affecting the system. Removing a redundant constraint can make other redundant constraints become

irredundant, so they usually they cannot be all removed at the same time. A system is irredundant if it has no redundant constraints.

Redundant constraints

Example 1:

c

c

c’

Constraint c can be removedwithout affecting the system

Example 2:

Constraint c and c’ are redundant and can beremoved without affecting the system, but notat the same time

Page 5: The Structure of Polyhedra

The characteristic/recession cone of a given polyhedron P, denoted by char.cone(P) is the polyhedral cone:

char.cone(P) = {y | x + y P for all x in P} = {y | Ay ≤ 0}

y char.cone(P) there is an x in P such that x + y P for all ≥ 0 P + char.cone(P) = P P is bounded char.cone(P) = {0} If P = Q + C, with Q a polytope and C a polyhedral cone, then C = char.cone(P) The nonzero vectors in char.cone(P) are called infinite directions of P

Characteristic/recession cone

Page 6: The Structure of Polyhedra

The lineality space of P, denoted my lin.space(P) is the linear space:

lin.space(P) = char.cone(P) – char.cone(P) = {y | Ay ≤ 0}

Lineality space

If lin.space(P) has dimension 0, then P is called pointed

Page 7: The Structure of Polyhedra

A nonempty polyhedron P can be uniquely represented as:

P = H + Q

where H is the lin.space(P), and Q is a nonempty pointed polyhedron.

The affine hull of P is given by:

affine.hull(P) = { x | A= x = b= } = { x | A= x ≤ b= }

If ax ≤ is an implicit equality in Ax ≤ b, the equality ax = is already

implied by A= x ≤ b=.

The dimension of P is equal to n – rank of matrix A=

P is full-dimensional if its dimension is n P is full-dimensional there are no implicit inequalities

Affine hull and dimension

Page 8: The Structure of Polyhedra

For a given set P, a hyperplane is called a supporting hyperplane if it contains P in one of its closed halfspaces and intersects the closure of P with at least one point

Supporting hyperplanes

Example 1:

Supporting hyperplanes Non-supporting hyperplanes

Example 2:

Page 9: The Structure of Polyhedra

A subset F of P is called a face of P if F = P or if F is the intersection of P with a supporting hyperplane of P (by convention is also a face).

Faces of dimension 0, 1,…, d – 2 and d – 1 are vertices, edges,…, ridges and facets respectively. Each face is a nonempty polyhedron.

Faces

Page 10: The Structure of Polyhedra

A facet of a convex polyhedral set P is a face of maximal dimension distinct from P (maximal relative to inclusion). If P is in Rd, its facets are the faces in Rd-1

If no inequality in A+x ≤ b+ is redundant in Ax ≤ b, then there exists a one-to-one correspondence between the facets of P and the inequalities in A+x ≤ b+. This implies: Each face of P, except for P itself, is the intersection of facets of P

P has no faces different from P P is an affine subspace The dimension of any facet in P is one less than the dimension of P

If P is full-dimensional, and Ax ≤ b is irredundant, then Ax ≤ b is the unique minimal representation of P.

Maximal faces: Facets

Intersection of 2 facets (R2)is an edge (R1)

Intersection of 3 facets (R2)is a vertex (R0)

Page 11: The Structure of Polyhedra

A minimal face of P is a face not containing any other faces. A face F of P is a minimal face F is an affine subspace Hoffman and Kruskal [1956]:

A set F is a minimal face of P ≠ F P and:

F = {x | A’x = b’}

for some subsystem A’x ≤ b’ in Ax ≤ b.

All faces of P have the same dimension, namely n minus the rank of A. If P is pointed, each minimal face consists of just one point. These points (or these minimal faces)

are called vertices of P. Each vertex is determined by n linearly independent equations from the system Ax = b. A vertex of {x | Ax ≤ b} is called a basic:

Feasible solution for Ax ≤ b. Optimum solution if it attains max {cx | Ax ≤ b} for some objective vector c.

Minimal faces: vertices

Page 12: The Structure of Polyhedra

The intersection of two faces is empty or a face again. Hence, the faces, together with form a lattice under inclusion, which is called face-lattice of P.

The face lattice

F1

F2

F3

F4

P

F1 F2 F3 F4

F1F4 F1F2 F2F3 F3F4

P

F1F4

F1F2

F2F3

F3F4

Fd(P)Polyhedron

Fd-1(P)Edges

F0(P)Vertices

F-1(P)Empty set

(-1)d

(-1)d-1

1

x (-1)

0

+

+

2D:

3D:

-1 + f0(P) – f1(P) + 1 = 0V = EV – E = 0-1 + V – E + 1 = 0

V – E + F = 2V – E + F – 2 = 0-1 + V – E + F – 1 = 0-1 + f0(P) – f1(P) + f2(P) – 1 = 0

Euler-Poincaré Relation

Page 13: The Structure of Polyhedra

Convex bounded polyhedra (polytopes) can be characterized as the convex hull of a set of points in some Rd.

The convex hull of a set of points P is the intersection of all convex sets containing P.

Convex hull

Points in P are either: Extreme points: vertices of the polytope. Interior points: they can be expressed as the convex combination of extreme points.

The convex combination of any two adjacent vertices (line segment) is an edge of P (face in dimension 1).

Page 14: The Structure of Polyhedra

Analog to vertices (extreme points), extremal rays cannot be represented as the non trivial convex combination of other rays.

Extremal rays

Example of extremal rays in R3

Page 15: The Structure of Polyhedra

Decomposition of polyhedra Any polyhedron has a unique minimal representation as:

P = conv.hull{x1,…,xn} + cone{y1,…,yn} + lin.space(P)

This is known as the “internal” representation, while the “external” representation is given by:

P = {x | A+x ≤ b+}

Page 16: The Structure of Polyhedra

Decomposition of polyhedra Any polyhedron has a unique minimal representation as:

P = conv.hull{x1,…,xn} + cone{y1,…,yn} + lin.space(P)

This is known as the “internal” representation, while the “external” representation is given by:

P = {x | A+x ≤ b+}

Page 17: The Structure of Polyhedra

Decomposition of polyhedra If P is convex and bounded (polytope), then its minimal

representation is given by:

P = conv.hull{x1,…,xn}

The set of points {x1,…,xn} are the extremal points (vertices – faces of dimension 0) of the polytope.

Page 18: The Structure of Polyhedra

Application Doubly stochastic matrices

A square matrix A = (ij)ni,j = 1 is called doubly stochastic if

Example of a doubly stochastic matrix: 1

1 ( 1,..., )n

iji

j n

1

1 ( 1,..., )n

ijj

i n

0 ( , 1,..., )ij i j n

0.3 0.1 0.6

0.5 0.1 0.4

0.2 0.8 0

Page 19: The Structure of Polyhedra

Application Permutation matrix

Matrix obtained by permuting the rows of an identity matrix according to some permutation of the numbers 1 to n. Every row and column contains precisely a single 1 with 0s everywhere else. There are n! permutation matrices of size n.

The permutation matrices for n = 2 are given by:

The permutation matrices for n = 3 are given by:

Page 20: The Structure of Polyhedra

Application Theorem – Birkhoff [1946] and von Newmann [1953]

Convex combination of:

Page 21: The Structure of Polyhedra

Application Theorem – Birkhoff [1946] and von Newmann [1953]

Convex combination of:

Permutation matrices of order 3

Page 22: The Structure of Polyhedra

Application Theorem – Birkhoff [1946] and von Newmann [1953]

Matrix A is doubly stochastic A is a convex combination of permutation matrices.

Proof: Sufficiency: direct as all permutation matrices are doubly stochastic.

Necessity: proved by induction on the order n of A (n = 1 trivially true)Consider the polytope P (in n2 dimension) of all doubly stochastic matrices of order n. P is defined by:

We have to show that each vertex of P is a permutation matrix.Let matrix A be a vertex of P, then n2 linearly independent constraints must besatisfied by A with equality.

1 1

1 ( , 1,..., )n n

ij iji j

i j n

0 ( , 1,..., )ij i j n

Page 23: The Structure of Polyhedra

Application Theorem – Birkhoff [1946] and von Newmann [1953]

Matrix A is doubly stochastic A is a convex combination of permutation matrices.

The first 2n constraints:

are linearly dependant. The last element of A is implied by the others, thus we have 2n – 1 constraints. Combining this with the n2 nonegativity constraints:

we then know that at least n2 – 2n + 1 of the ij are 0. This is, there is a row in A whereall elements are 0 except for one of them. Since the matrix is doubly stochastic, thenthat element must be 1.

Without loss of generality, suppose such element is a11 = 1, then removing first row andcolumn gives a doubly stochastic matrix of order n – 1, which by induction hypothesis isthe convex combination of permutation matrices (doubly stochastic matrix).

1 1

1 ( , 1,..., )n n

ij iji j

i j n

0 ( , 1,..., )ij i j n

Page 24: The Structure of Polyhedra

Application Corollary – Frobenius [1912, 1917]

Each regular bipartite graph G of degree r ≥ 1 has a perfect matching. A bipartite graph, also called a bigraph, is a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent A matching on a graph is a set of edges of such that no two of them share a vertex in common. The largest possible matching on a graph with n nodes consists of n/2 edges, and such a matching is called a perfect matching.

Let {1,…,n} and {n + 1,…,2n} be the color classes of G, and consider the

n-by-n matrix A = (ij) with

ij := 1/r (number of edges connecting i and n + j)

Then A is doubly stochastic, hence there is a permutation matrix B = (ij)

such that ij > 0 if = 1. Matrix B gives a perfect matching in G.

Page 25: The Structure of Polyhedra

Application Corollary – Frobenius [1912, 1917]

Each regular bipartite graph G of degree r ≥ 1 has a perfect matching.

1

2

3

5

6

7

4 8

5 6 7 8

1 1/3 1/3 1/3 0

2 0 1/3 1/3 1/3

3 1/3 0 1/3 1/3

4 1/3 1/3 0 1/3

Page 26: The Structure of Polyhedra

Application Corollary – Frobenius [1912, 1917]

Each regular bipartite graph G of degree r ≥ 1 has a perfect matching.

1

2

3

5

6

7

4 8

5 6 7 8

1 1/3 1/3 1/3 0

2 0 1/3 1/3 1/3

3 1/3 0 1/3 1/3

4 1/3 1/3 0 1/3

5 6 7 8

1 1 0 0 0

2 0 1 0 0

3 0 0 1 0

4 0 0 0 1

5 6 7 8

1 0 1 0 0

2 0 0 1 0

3 0 0 0 1

4 1 0 0 0

Page 27: The Structure of Polyhedra

Application Corollary – Frobenius [1912, 1917]

Perfect matching polytope

Let G = (V, E) be an undirected graph. The perfect marching polytope of G is the convex hull of the characteristic vectors of perfect matchings of G. So P is a polytope in RE. Each vector x in the perfect matching polytope satisfies:

Where (v) denotes the set of edges incident with v. In other words, if

a graph G is bipartite, then the perfect matching polytope is completely

defined by this system.

Notation1

5

2

3

4

x = [1 0 1 0 1]1 2 3 4 5

( )

0

1e

ee v

x

x

( )

( )

e E

v V

Page 28: The Structure of Polyhedra

Application Corollary – Frobenius [1912, 1917]

ProofEach vector satisfying

is a convex combination of characteristic vectors of perfect matching in G, if Gis bipartite. Let x satisfy these constraints and V1 = {1,…,n} and V2 = {n + 1,…,2n}be the two color classes of G. Then:

Let A = (aij) be the n-by-n matrix defined by:

aij := 0 if {I, n + j} Eaij := xe if {I, n + j} E

Then A is doubly stochastic: convex combination of permutation matrices, each ofThem corresponding to a perfect matching in G. Thus x is a convex combinationof characteristic vectors of perfect matchings in G.

( )

0

1e

ee v

x

x

( )

( )

e E

v V

1 2

1 2( ( )) ( ( ))v V v V

V x v x v V

Page 29: The Structure of Polyhedra

Application The matching polytope

Generalization to general, not necessarily bipartite graphs.Let G = (V, E) be an undirected graph, with |V| even, and let P be the

|E|-dimensional associated perfect matching polytope (convex hull of the

characteristic vectors of the prefect matchings in G).

For non-bipartite graphs, the constraints:

Are not enough to determine P. Proof (by counter example)

( )

0

1e

ee v

x

x

( )

( )

e E

v V

x = [½ ½ ½ ½ ½ ½]

Page 30: The Structure of Polyhedra

Application The matching polytope

Edmonds’ matching polyhedron theorem [1965]The perfect matching polytope for general, not necessarily bipartite graphs, is

given by:

where (W) is the set of edges of G intersecting W in exactly one vertex.

( )

( )

1

1

ee v

ee w

x

x

( )e E0ex ( )v V

W odd

Page 31: The Structure of Polyhedra

Thank you !