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.

翻译：整数规划（IP）问题中，若其整数系数矩阵$M$的所有子式绝对值均受常数$Δ$限制，则是否存在多项式时间求解算法是一个众所周知的开放难题。本文在进一步要求从$M$中移除常数行与常数列后，所得子矩阵$A$为网络矩阵的转置时，对该问题给出了肯定回答。我们的研究方法集中于从$M$中仅移除$k$行（$k$为常数）得到$A$的情形。研究结果通过两个主要步骤实现：第一步涉及整数规划理论，第二步涉及图子式理论。首先，针对$A$为一般全幺模矩阵的情形，我们推导出一个强邻近性结论：给定线性规划松弛的最优解，通过寻找$[A\; I]$的回路进行常数次增广即可获得整数规划的最优解。其次，针对$A$为网络矩阵转置的情形，我们将问题重新表述为图$G$上的最大约束整数势问题。我们观察到若$G$是$2$-连通的，则当$t = Ω(k Δ)$时不存在根化$K_{2,t}$-子式。利用此性质，我们将$G$树分解为高度结构化的子图，从而在局部求解问题。最终通过动态规划方法实现全局问题的求解。