Let $A \in \mathbb{Z}^{m \times n}$ be an integral matrix and $a$, $b$, $c \in \mathbb{Z}$ satisfy $a \geq b \geq c \geq 0$. The question is to recognize whether $A$ is $\{a,b,c\}$-modular, i.e., whether the set of $n \times n$ 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 $n \times n$ subdeterminants $k_1$ and $k_2$ satisfying $2 \cdot |k_1| = |k_2|$. 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 $c$.


翻译:问题在于确认美元是否为$a, b, c $-modual, 即绝对值为$n n un minutes 的一套美元绝对值为$a, b, c 美元, c 美元 美元。我们将在多元时间成功地解决这个问题,除非美元具有重复关系,即美元具有非零 美元 美元 = 美元 = geq b, c = geq c = geq = 0 美元。 问题在于确认美元是否为$a, b, c = 美元 美元 = 美元 = $k_ 美元 = = k_ 2 美元。这是我们分析的结果,我们为任何完全单一的矩阵提供了一种众所周知的识别算法, 美元 美元 = 美元 美元 = 美元 = 美元 = 美元 = = 美元 = 美元 = 2 = = = 美元 = = 美元 = = = 美元 = = = = 任何标准的矩阵 美元 = 美元 = =xxxxxxxxxxxxxxxx y y y = = 美元 = 美元 = 美元 = = = = = = = = = = = = = = = = = 美元 = = = = = = = = = = = = = = = = 美元 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

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