We introduce the tree-decomposition-based parameter totally $Δ$-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth in polynomial time when variables have bounded domain. This extends previous graph-based decomposition parameters for matrices with at most two nonzero entries per row to include matrices with entries outside of $\{-1,0,1\}$. We also give an analogue of the Grid Theorem of Robertson and Seymour for matrices of bounded TDM-treewidth in the language of rooted signed graphs.

翻译：我们引入了基于树分解的参数——完全$Δ$-模树宽（TDM-treewidth），该参数适用于每行有两个非零元素的矩阵。我们证明了当变量具有有界定义域时，具有有界TDM-treewidth的矩阵的整数规划可以在多项式时间内求解。这将对每行最多有两个非零元素的矩阵的基于图的分解参数进行了推广，使其能够包含元素不在$\{-1,0,1\}$中的矩阵。我们还给出了Robertson和Seymour的格定理在带根符号图语言下对于有界TDM-treewidth矩阵的类似物。