🤖 AI Summary
This work addresses the computational bottleneck in solving 4-block integer linear programs with large coefficients by introducing a novel approach based on an affine decomposition of the right-hand side. The method extends the existing n-fold integer programming algorithm to settings involving global variables. By leveraging key technical ingredients—including a vector rearrangement lemma, face guessing in hyperplane arrangements, dynamic high-multiplicity encoding, and faithful decomposition—the algorithm achieves, for the first time, polynomial dependence on the coefficient matrices A, B, and C. Consequently, the running time is significantly improved from the previous bound of \(n^{k^2 + O(1)}\) to \(f(k, \bar{\Delta}) \cdot n^{k + O(1)}\), where \(f\) depends only on the number of blocks \(k\) and the maximum subdeterminant \(\bar{\Delta}\), substantially outperforming the best-known prior results.
📝 Abstract
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 $\overline\Delta$ in absolute value. Our algorithm runs in time $f(k,\overline\Delta)\cdot n^{k+\mathcal O(1)}$, improving upon the previous best running time of $f(k,\overline\Delta)\cdot n^{k^2+\mathcal O(1)}$ [Oertel, Paat, and Weismantel (Math. Prog. 2024), Chen, Kouteck\'y, 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, Kouteck\'y, 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.