具有有界子行列式且每行两个非零项的整数规划
Integer programs with bounded subdeterminants and two nonzeros per row
AI总结：
本文提出了一种针对每行最多两个非零项且子行列式有界的整数线性规划的强多项式时间算法，其核心是首个解决不含超过 $k$ 个顶点不相交奇圈图的加权稳定集问题的多项式时间算法。
摘要：
我们为整数线性规划给出了一个强多项式时间算法，该规划由子行列式受常数界且每行最多含两个非零项的整数系数矩阵定义。我们方法的核心是首个针对不含超过 $k$ 个顶点不相交奇圈的图的加权稳定集问题的多项式时间算法，其中 $k$ 为任意常数。此前，仅知 $k=0$（二部图）和 $k=1$ 时存在多项式时间算法。我们观察到，由子行列式有界且每列两个非零项的系数矩阵定义的整数线性规划，也可通过归约至 $b$-匹配在强多项式时间内求解。
英文摘要：
We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than $k$ vertex-disjoint odd cycles, where $k$ is any constant. Previously, polynomial-time algorithms were only known for $k=0$ (bipartite graphs) and for $k=1$. We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to $b$-matching.
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