The Structure of Polyhedra

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The Structure of Polyhedra
CAS 746 Advanced Topics in Combinatorial
Optimization

• Gabriel Indik
• March 2006

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Presentation outline

• 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

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Implicit equalities

• 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

Example Take the following system of
inequalities a, b, c and d in R2
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.
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Redundant constraints

• 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.

Example 2
Example 1
Constraint c and c are redundant and can
be removed without affecting the system, but
not at the same time
Constraint c can be removed without affecting the
system
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Characteristic/recession cone

• 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

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Lineality space

• 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

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

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Affine hull and dimension

• 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

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Supporting hyperplanes

• 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

Example 2
Example 1
Non-supporting hyperplanes
Supporting hyperplanes
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Faces

• 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.

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Maximal faces Facets

• 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 Ax b is redundant in Ax
  b, then there exists a one-to-one correspondence
  between the facets of P and the inequalities in
  Ax 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.

Intersection of 3 facets (R2) is a vertex (R0)
Intersection of 2 facets (R2) is an edge (R1)
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Minimal faces vertices

• 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 Ax b
• for some subsystem
  Ax 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.

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The face lattice

• 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.

F1?F2
F2
F2?F3
F1
F3
F1?F4
F4
F3?F4
Euler-Poincaré Relation
-1 f0(P) f1(P) 1 0
V E
V E 0
-1 V E 1 0
2D 3D
V E F 2
V E F 2 0
-1 V E F 1 0
-1 f0(P) f1(P) f2(P) 1 0
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Convex hull

• 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.

• 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).

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Extremal rays

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

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Decomposition of polyhedra

• Any polyhedron has a unique minimal
  representation as
• P conv.hullx1,,xn coney1,,yn
  lin.space(P)
• This is known as the internal representation,
  while the external representation is given by
• P x Ax b

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Decomposition of polyhedra

• Any polyhedron has a unique minimal
  representation as
• P conv.hullx1,,xn coney1,,yn
  lin.space(P)
• This is known as the internal representation,
  while the external representation is given by
• P x Ax b

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Decomposition of polyhedra

• If P is convex and bounded (polytope), then its
  minimal representation is given by
• P conv.hullx1,,xn

• The set of points x1,,xn are the extremal
  points (vertices faces of dimension 0) of the
  polytope.

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Application

• Doubly stochastic matrices
• A square matrix A (?ij)ni,j 1 is called
  doubly stochastic if
• Example of a doubly stochastic matrix

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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

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Application

• Theorem Birkhoff 1946 and von Newmann 1953

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Application

• Theorem Birkhoff 1946 and von Newmann 1953

Convex combination of
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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 be
• satisfied by A with equality.

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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 where
• all elements are 0 except for one of them. Since
  the matrix is doubly stochastic, then
• that element must be 1.
• Without loss of generality, suppose such element
  is a11 1, then removing first row and
• column gives a doubly stochastic matrix of order
  n 1, which by induction hypothesis is
• the convex combination of permutation matrices
  (doubly stochastic matrix).

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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 gt 0 if ? 1. Matrix B gives a
  perfect matching in G.

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Application

• Corollary Frobenius 1912, 1917
• Each regular bipartite graph G of degree r 1
  has a perfect matching.

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Application

• Corollary Frobenius 1912, 1917
• Each regular bipartite graph G of degree r 1
  has a perfect matching.

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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.

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Application

• Corollary Frobenius 1912, 1917
• Proof
• Each vector satisfying
• is a convex combination of characteristic vectors
  of perfect matching in G, if G
• is 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 ?? E
• aij xe if I, n j ?? E

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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)

x ?½ ½ ½ ½ ½ ½
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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.

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Thank you !