Notes on {a,b,c}-Modular Matrices
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AbstractLet $A \in \mathbb {Z}^{m \times n}$ A ∈ ℤ m × n be an integral matrix and a, b, $c \in \mathbb {Z}$ c ∈ ℤ satisfy a ≥ b ≥ c ≥ 0. The question is to recognize whether A is {a,b,c}-modular, i.e., whether the set of n × n subdeterminants of A in absolute value is {a,b,c}. We will succeed in solving this problem in polynomial time unless A possesses a duplicative relation, that is, A has nonzero n × n subdeterminants k1 and k2 satisfying 2 ⋅|k1| = |k2|. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over {a,b,c}-modular constraint matrices for any constants a, b and c.
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2014 ◽
Vol 61
(1)
◽
pp. 51-78
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2002 ◽
Vol 50
(8)
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pp. 1935-1941
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