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

2. 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

3. 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 R 2 : 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

4. 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 removed without affecting the system Example 2: Constraint c and c’ are redundant and can be removed without affecting the system, but not at the same time

5. 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

6. 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

7. 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

8. 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:

9. 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

10. 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 R d, its facets are the faces in R d-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 (R 2 ) is an edge (R 1 ) Intersection of 3 facets (R 2 ) is a vertex (R 0 )

11. 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

12. 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 F1F1 F2F2 F3F3 F4F4 P F1F1 F2F2 F3F3 F4F4 F1F4F1F4 F1F2F1F2 F2F3F2F3 F3F4F3F4 P  F1F4F1F4 F1F2F1F2 F2F3F2F3 F3F4F3F4 F d (P) Polyhedron F d-1 (P) Edges F 0 (P) Vertices F -1 (P) Empty set  (-1) d  (-1) d-1  1 x (-1) 0 + + 2D: 3D: -1 + f 0 (P) – f 1 (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 + f 0 (P) – f 1 (P) + f 2 (P) – 1 = 0 Euler-Poincaré Relation

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

14. 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 R 3

15. Decomposition of polyhedra Any polyhedron has a unique minimal representation as: P = conv.hull{x 1,…,x n } + cone{y 1,…,y n } + lin.space(P) This is known as the “internal” representation, while the “external” representation is given by: P = {x | A + x ≤ b + }

16. Decomposition of polyhedra Any polyhedron has a unique minimal representation as: P = conv.hull{x 1,…,x n } + cone{y 1,…,y n } + lin.space(P) This is known as the “internal” representation, while the “external” representation is given by: P = {x | A + x ≤ b + }

17. Decomposition of polyhedra If P is convex and bounded (polytope), then its minimal representation is given by: P = conv.hull{x 1,…,x n } The set of points {x 1,…,x n } are the extremal points (vertices – faces of dimension 0) of the polytope.

18. Application Doubly stochastic matrices  A square matrix A = (  ij ) n i,j = 1 is called doubly stochastic if  Example of a doubly stochastic matrix: 0.30.10.6 0.50.10.4 0.20.80

19. 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:

20. Application Theorem – Birkhoff [1946] and von Newmann [1953] Convex combination of:

21. Application Theorem – Birkhoff [1946] and von Newmann [1953] Convex combination of: Permutation matrices of order 3

22. 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 n 2 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.vertex Let matrix A be a vertex of P, then n 2 linearly independent constraints must be satisfied by A with equality.

23. 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 n 2 nonegativity constraints: we then know that at least n 2 – 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 a 11 = 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).

24. 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.

25. 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 5678 1 1/3 0 2 0 3 0 4 0

26. 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 5678 1 1/3 0 2 0 3 0 4 0 5678 1 1000 2 0100 3 0010 4 0001 5678 1 0100 2 0010 3 0001 4 1000

27. 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 R E. 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. Notation 1 5 2 3 4 x = [1 0 1 0 1] 1 2 3 4 5

28. 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 V 1 = {1,…,n} and V 2 = {n + 1,…,2n} be the two color classes of G. Then: Let A = (a ij ) be the n-by-n matrix defined by: a ij := 0 if {I, n + j}  E a ij := x e if {I, n + j}  E Then A is doubly stochastic: convex combination of permutation matrices, each of Them corresponding to a perfect matching in G. Thus x is a convex combination of characteristic vectors of perfect matchings in G.

29. 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]

30. 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.

31. Thank you !