We present a new and faster algorithm for the 4-block integer linear programming problem, overcoming the long-standing runtime barrier faced by previous algorithms that rely on Graver complexity or proximity bounds. The 4-block integer linear programming problem asks to compute $\min\{c_0^\top x_0+c_1^\top x_1+\dots+c_n^\top x_n\ \vert\ Ax_0+Bx_1+\dots+Bx_n=b_0,\ Cx_0+Dx_i=b_i\ \forall i\in[n],\ (x_0,x_1,\dots,x_n)\in\mathbb Z_{\ge0}^{(1+n)k}\}$ for some $k\times k$ matrices $A,B,C,D$ with coefficients bounded by $\overlineΔ$ in absolute value. Our algorithm runs in time $f(k,\overlineΔ)\cdot n^{k+\mathcal O(1)}$, improving upon the previous best running time of $f(k,\overlineΔ)\cdot n^{k^2+\mathcal O(1)}$ [Oertel, Paat, and Weismantel (Math. Prog. 2024), Chen, Koutecký, Xu, and Shi (ESA 2020)]. Further, we give the first algorithm that can handle large coefficients in $A, B$ and $C$, that is, it has a running time that depends only polynomially on the encoding length of these coefficients. We obtain these results by extending the $n$-fold integer linear programming algorithm of Cslovjecsek, Koutecký, Lassota, Pilipczuk, and Polak (SODA 2024) to incorporate additional global variables $x_0$. The central technical result is showing that the exhaustive use of the vector rearrangement lemma of Cslovjecsek, Eisenbrand, Pilipczuk, Venzin, and Weismantel (ESA 2021) can be made \emph{affine} by carefully guessing both the residue of the global variables modulo a large modulus and a face in a suitable hyperplane arrangement among a sufficiently small number of candidates. This facilitates a dynamic high-multiplicy encoding of a \emph{faithfully decomposed} $n$-fold ILP with bounded right-hand sides, which we can solve efficiently for each such guess.

翻译：本文针对四块整数线性规划问题提出了一种新的快速算法，突破了以往依赖Graver复杂度或邻近性界限的算法长期面临的时间复杂度障碍。四块整数线性规划问题要求计算 $\min\{c_0^\top x_0+c_1^\top x_1+\dots+c_n^\top x_n\ \vert\ Ax_0+Bx_1+\dots+Bx_n=b_0,\ Cx_0+Dx_i=b_i\ \forall i\in[n],\ (x_0,x_1,\dots,x_n)\in\mathbb Z_{\ge0}^{(1+n)k}\}$，其中 $A,B,C,D$ 为 $k\times k$ 矩阵且系数绝对值以 $\overlineΔ$ 为界。我们的算法运行时间为 $f(k,\overlineΔ)\cdot n^{k+\mathcal O(1)}$，较先前最佳运行时间 $f(k,\overlineΔ)\cdot n^{k^2+\mathcal O(1)}$ [Oertel, Paat, and Weismantel (Math. Prog. 2024), Chen, Koutecký, Xu, and Shi (ESA 2020)] 有显著改进。此外，我们首次给出了能够处理 $A, B, C$ 中大幅值系数的算法，其运行时间仅与这些系数编码长度的多项式相关。这些成果是通过扩展 Cslovjecsek, Koutecký, Lassota, Pilipczuk, and Polak (SODA 2024) 提出的 $n$-折叠整数线性规划算法，使其能够纳入额外的全局变量 $x_0$ 而实现的。核心技术贡献在于证明：通过精心猜测全局变量在较大模数下的剩余值以及合适超平面排列中的一个面（候选数量足够有限），可以使得 Cslovjecsek, Eisenbrand, Pilipczuk, Venzin, and Weismantel (ESA 2021) 向量重排引理的穷举使用具有仿射特性。这为具有有界右端项的忠实分解 $n$-折叠整数线性规划问题实现了动态高重数编码，使得我们能够对每个此类猜测进行高效求解。