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Integer program with bimodular matrix

By S. I. Veselov and A. J. Chirkov


We use the following notation: if A is a matrix, Ai denotes its i- th row, ai,j is the entry at row i and column j, ∆k(A) is the maximum of the absolute values of the k × k minors of A; if b is a vector then bi denotes its i- th coordinate. Let ⌊α ⌋ denote the largest integer less than or equal to α. The transpose of a matrix A is denoted by A T. We say that A ∈ Z m×n is bimodular if rank A = n and ∆n(A) ≤ 2. By definition, put SZ = conv(S ∩ Z n) for every S ⊆ R n. Let M(A, b) be the set {x ∈ R n: Ax ≤ b}. Theorem 1 If A is bimodular and M(A, b) is full-dimensional, then MZ(A, b) is non-empty. Proof. We prove the statement by induction on n. If n = 1, then M(A, b) ⊇ {x ⌋ ∈ R: β − 1 ≤ αx ≤ β} for some β ∈ Z and α ∈ {1, 2}. It is clear that ∈ MZ(A, b)

Year: 2008
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