具有近完全单模矩阵的整数规划：连通性情况
Integer programs with nearly totally unimodular matrices: the cographic case
AI总结：
该研究证明了当整数规划的系数矩阵经过移除常数个行和列后成为网络矩阵转置时，可以通过树分解和动态规划在多项式时间内求解。
摘要：
对于具有整数系数矩阵M的整数规划（IPs），其中所有子行列式都绝对值不超过常数Δ，是否可以在多项式时间内求解是一个著名的开放问题。如果我们进一步要求，通过移除M中的常数个行和列后，得到的子矩阵A是网络矩阵的转置，那么我们可以肯定地回答这个问题。我们的方法聚焦于A由移除k行（k为常数）后从M得到的情况。我们通过两个主要步骤实现了这一结果，第一个与整数规划的理论有关，第二个与图 minors 理论有关。首先，我们推导出当A是通用完全单模矩阵时的强接近结果：给定线性规划松弛的最优解，可以通过寻找[A I]的常数个电路的增益来获得整数规划的最优解。其次，对于A是网络矩阵转置的情况，我们将问题重新表述为图G上的最大受限整数势问题。我们观察到，如果G是2-连通的，那么它没有根K_{2,t}-minor，其中t=Ω(kΔ)。我们利用这一点，得到G的树分解，将其分解为高度结构化的图，可以局部解决该问题。这使得我们能够通过动态规划解决全局问题。
英文摘要：
It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix $M$ whose subdeterminants are all bounded by a constant $Δ$ in absolute value, can be solved in polynomial time. We answer this question in the affirmative if we further require that, by removing a constant number of rows and columns from $M$, one obtains a submatrix $A$ that is the transpose of a network matrix. Our approach focuses on the case where $A$ arises from $M$ after removing $k$ rows only, where $k$ is a constant. We achieve our result in two main steps, the first related to the theory of IPs and the second related to graph minor theory. First, we derive a strong proximity result for the case where $A$ is a general totally unimodular matrix: Given an optimal solution of the linear programming relaxation, an optimal solution to the IP can be obtained by finding a constant number of augmentations by circuits of $[A\; I]$. Second, for the case where $A$ is transpose of a network matrix, we reformulate the problem as a maximum constrained integer potential problem on a graph $G$. We observe that if $G$ is $2$-connected, then it has no rooted $K_{2,t}$-minor for $t = Ω(k Δ)$. We leverage this to obtain a tree-decomposition of $G$ into highly structured graphs for which we can solve the problem locally. This allows us to solve the global problem via dynamic programming.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
