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 when variables are bounded. 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：本文针对每行恰有两个非零元素的矩阵，提出了基于树分解的参数——完全Δ-模树宽（TDM-treewidth）。我们证明了当变量有界时，如何求解具有有界TDM-treewidth矩阵的整数规划问题。这一结果将先前针对每行至多两个非零元素矩阵的基于图的分解参数，推广到包含元素超出$\{-1,0,1\}$范围的矩阵。此外，我们以有根符号图的语言，给出了Robertson与Seymour网格定理在具有有界TDM-treewidth矩阵上的类比形式。