Integer Programming and Totally unimodular matrices

Integer Programming and Totally

unimodular matrices

Carlo Mannino

(from Geir Dahl and Carlo Mannino notes)

Integer Programming

max {cTx: xP  Zn }

P_I = conv(P  Zn ) convex hull of integer points in P is a polyhedron

max {cTx: xP  Zn } = max {cTx: x  P_I}  max {cTx: x  P} = UB

linear relaxation P  Rn polyhedron Integer Programming P P P_I P is called integral if P = PI

Characterizing integer polyhedra

Let P  Rn _{be a non-empty polyhedron. The following} statements are equivalent:

(i) P is integral.

(ii) Each minimal face of P contains an integral vector.

(iii) max {cTx: xP} is attained for an integral vector for each cRn for which the maximum is finite.

Furthermore, if P is pointed, then P is integral if and only if each vertex is integral.

Exposed faces

asda

Let P = {xRn : Ax ≤ b} be a polyhedron, with ARm,n . A

non-empty set F is an exposed face (=face) of P if and only if F = {xP: A’x = b’ }

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

Proposition :

The following is an important fact (Proposition 4.4.6 from Geir Dahl’s notes An introduction to convexity)

Totally unimodular matrices

ARm,n is totally unimodular (TU) if the determinant of each square submatrix is either -1, 0 or 1.

Obs: every element of A must be -1, 0 or 1

Let ARm,n be TU and let bRm be an integral vector. Then the polyhedron P = {x Rn : Ax  b} is integral.

Proposition 2.5

Cramer’s Rule to compute the inverse matrix : C non-singular square matrix

C_ij obtained from C by deleting row i and column j

C-1 inverse of C  (j,i) element of C-1 = (-1)i+j det(C_ij)/det(C)

Proof of Theorem 2.5

Let F={x Rn _{: A}’_{x = b}’_{} be a minimal face of P (where {A}’_x  _b’ _} is a subsystem of {Ax  b })

We can assume the rows in {A’x = b’} be linearly independent.

Then A’ contains a m’m’ nonsingular submatrix B such that A’= [B N] x F, then x = [x_B , x_N]  x_B = B-1b’, x

N = 0.

B-1 and b’ integers  x_B integer  F contains an integer point  P integer polyhedron

Preserving total unimodularity

Matrix operations preserving total unimodularity

transpose

augmenting with the identity matrix multiplying a column or a row by -1

interchanging two rows or two columns duplication of rows and columns

Totally unimodular matrices and duality

Let ARm,n be TU and bRm, cRn integral vectors. Then each of the dual LP problems in the dual optimal relation: max{cTx: Ax  b} = min{bTy: ATy=c, y0}

has an integral optimal solution

Characterizing totally unimodular matrices

A{0,1,-1}m,n is TU iff for each I{1,…,I} there is a partition I₁, I₂ of I such that

Theorem 2.9: Ghouila-Houri characterization

. , , 1 for 1 2 1 n j a a I i ij I i ij 





Since the transpose of a TU matrix is also TU, the Ghouila-Houri property holds for columns as well.

Node-edge incidence matrices

G = (V,E) undirected graph.

A_GRV,E node-edge incidence matrix of G e₅ 1 5 2 3 e₁ 4 e₂ e₃ e₄                 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 e₁ e₂ e₃ e₄ e₅ 1 2 3 4 5

A_G is TU if and only if G is bipartite.

Proposition 2.10:

G = (V,E) is bipartite if there is a partition I₁ ,I₂ of V such that every edge has one endpoint in I₁ and the other endpoint in I₂

column  edge

Proof of proposition 2.12

Let I  V be a subset of rows of A_G. Let I’₁ = I  I₁ and I’₂ = I  I₂

e₅ 1 5 2 3 e₁ 4 e₂ e₃ e₄                 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 e₁ e₂ e₃ e₄ e₅ 1 2 3 4 5 I’₁ I’₂ + - - = 1 0 0 -1 -1 Let a_e be the sum associated to e = (u,v)

.

|

,|

,

1

for

1

E

i

a

a













Then a_e = 0 if u  I’₁ and v  I’₂ ae = 1 if u  I’1 and v  I

a_e = -1 if u  I and v  I’₂ Proof. If

Proof of proposition 2.12

If G non bipartite G contains an odd cycle C (show it!) A_C submatrix of A_G associated to nodes and edges of C Proof. Only If 1 3 2 5 4                 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 2 3 4 5 A_C circulant matrix (after reordering)

Some interesting dual pairs

Let G be a bipartite graph A_G its incidence matrix. Then each of the dual problems in the dual optimal relation: max{1Tx: A_Gx  1} = min{yT1: yTA_G 1, y0}

have integer optimal solution

Corollary

The above linear programs have a nice combinatorial interpretation The primal corresponds with finding a maximum cardinality matching

The dual corresponds with finding a minimum cardinality node cover

Matching

G = (V,E) undirected graph.

Matching M  E: subset of edges meeting each node at most once x{0,1}E _{incidence vector of matching in G}  _x(



e₅ 1 5 2 3 e₁ 4 e₂ e₃ e₄ x₁ + x₂ + x₃  1 x₄ + x₅  1











Maximum Cardinality Matching = max{1Tx: A_Gx  1, x {0,1}E}

max{1Tx: A_Gx  1, x  0} LP!

Node Cover

G = (V,E) undirected graph.

Node cover C: subset of nodes meeting each edge at least once

y{0,1}V incidence vector of node cover in G  y_u + y_v  1 for all uv E

e₅ 1 5 2 3 e₁ 4 e₂ e₃ e₄ y₁ + y₃  1 y₁ + y₄  1 e₁ e₂ y₁ + y₅  1 y₂ + y₄  1 e₃ e₄ y₂ + y₅  1 e₅ yTA_G  1

Minimum Cardinality Node Cover = min{yT1: yTA_G 1, y {0,1}V } min{yT1: yTA_G 1, y0} LP!

Some interesting dual pairs

The primal is the maximum cardinality node packing (stable set)

The dual is the minimum cardinality edge cover.

The cardinalities of such sets coincide in bipartite graphs (Konig’s covering theorem).

Let G be a bipartite graph A_G its incidence matrix. Then each dual problems in the dual optimal relation:

have integer optimal solution

Corollary } 0 , 1 : 1 min{ } 0 , 1 : 1 max{ T y A_GTy  y   xT xTA_GT  T x 

Edge Cover

G = (V,E) undirected graph.

Edge Cover C  E: subset of edges meeting each node at least once x{0,1}E _{incidence vector of edge cover in G}  _x(



e₅ 1 5 2 3 e₁ 4 e₂ e₃ e₄ x₁ + x₂ + x₃  1 x₄ + x₅  1











Minimum Cardinality Edge Cover =

LP! G bipartite  A_G is TU

=

Node Packing

G = (V,E) undirected graph.

Node packing S: subset of nodes meeting each edge at most once

y{0,1}V incidence vector node packing in G  y_u + y_v 1 for all uv E

e₅ 1 5 2 3 e₁ 4 e₂ e₃ e₄ y₁ + y₃  1 y₁ + y₄  1 e₁ e₂ y₁ + y₅  1 y₂ + y₄  1 e₃ e₄ y₂ + y₅  1 e₅

Maximum Cardinality Node Packing =

LP! G bipartite  A_G is TU _{max{1 y :}

=

Node-arc incidence matrices

D = (V,E) directed graph.

A_GRV,E node-edge incidence matrix of G

A_D is TU for any directed graph D.

Proposition 2.12:

column  arc row  node star

s 1 2 _t 3 e₃ e₂ e₁ e₇ e₄ e₆ e₅                        1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 s 2 3 4 t e₁ e₂ e₃ e₄ e₅ e₆ e₇