IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

IEOR269 notes, Prof. Hochbaum, 2010 3
otherwise. This is illustrated for the case of n = 3 in equation (3), where the columns of A are
ordered in lexicographic order for (i, j) ∈ {1, . . . , n} 2 .
⎛
⎞
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
A =
0 0 0 0 0 0 1 1 1
⎜1 0 0 1 0 0 1 0 0
⎟
⎝0 1 0 0 1 0 0 1 0⎠
0 0 1 0 0 1 0 0 1
and if we note b = ( 1, 1, 1, 1, 1, 1, 1, 1, 1 ) T and X =
(
x1,1 , x 1,2 , x 1,3 , . . . , x 3,1 , x 3,2 , x 3,3
) T the constraints
(1)- (2) in the case of n = 3 can be rewritten as AX = b.
This property of the constraint matrix can be used to show that the solutions of the LPrelaxation
of the assignment problem are integers.
Definition 2.1. A matrix A is called totally unimodular (TUM) if for any square sub-matrix A ′
of A, det A ′ ∈ {−1, 0, 1}.
Lemma 2.2. The solutions of a LP with integer objective, integer right-hand side constraint, and
TUM constraint matrix are integers.
Proof. Consider a LP in classical form:
max c T x
subject to A x ≤ b
where c, x ∈ Z n , A ∈ Z m×n totally unimodular, b ∈ Z m . If m < n, a basic solution x ∗ basic
of the
LP is given by x ∗ basic = B−1 b where B = (b i,j ) 1≤i,j≤m is a non-singular square submatrix of A
(columns of B are not necessarily consecutive columns of A).
Using Cramer’s rule, we can write: B −1 = C T / det B where C = (c i,j ) denotes the cofactor
matrix of B. As detailed in (4), the absolute value of c i,j is the determinant of the matrix obtained
by removing line i and column j from B.
⎛
⎞
b 1,1 . . . b 1,j−1 X b 1,j+1 . . . b 1,m
. . . . . . .
c i,j = (−1) i+j b i−1,1 . . . b i−1,j−1 X b i−1,j+1 . . . b i−1,m
det
X X X X X X X
(4)
b i+1,1 . . . b i+1,j−1 X b i+1,j+1 . . . b i+1,m
⎜
⎟
⎝ . . . . . . . ⎠
b m,1 . . . b m,j−1 X b m,j+1 . . . b m,m
Since det(·) is a polynomial function of the matrix coefficients, if B has integer coefficients, so does
C. If A is TUM, since B is non-singular, det(B) ∈ {−1, 1} so B −1 has integer coefficients. If b has
integer coefficients, so does the optimal solution of the LP.
ILP with TUM constraint matrix can be solved by considering their LP relaxation, whose solutions
are integral. The minimum cost network flow problem has a TUM constraint matrix.
(3)