Solving 4-Block Integer Linear Programs Faster Using Affine Decompositions of the Right-Hand Sides

TL;DR

This work breaks new ground for 4-block ILPs by introducing an affine, region-wise decomposition framework that extends the $n$-fold ILP approach. It achieves a faster slice-wise running time $f(k,\overline{\Delta})\cdot n^{k+\mathcal{O}(1)}$ and, importantly, provides the first algorithm with polynomial dependence on the encoding length of large coefficients, thanks to affine faithful decompositions and dynamic handling of the RHS via hyperplane arrangements. The key technical contribution is showing that the vector rearrangement lemma can be made affine, enabling high-multiplicity encodings of faithfully decomposed $n$-fold ILPs with bounded RHSs and efficient per-region solves, which leads to improved exponents in $n$ and a path toward coefficient-insensitive performance. Overall, the paper advances the understanding of 4-block ILP tractability and offers a scalable, region-based framework that could influence future FPT investigations and extensions to related block-structured ILPs.

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 $\overlineΔ$ 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.

Figures (3)

• Figure 1: Displayed are two prominent special cases of 4-block matrices.
• Figure 4: A partition of $\mathbb Z^2$ into four remainder classes modulo $2$, indicated by the differently shaped markers placed at each lattice point, and faces induced by $5$ hyperplanes, indicated by differently colored regions for the $2$-dimensional faces. Note that lower-dimensional faces also appear in arrangements and may contain integral points. To solve a 4-block ILP, we solve a configuration ILP for each set of identically shaped and colored points and restricting $x_0$ to be in such set.
• Figure 5: Visual representation of a subregion of $\mathbb R^2$ in the proof of \ref{['lemma:dynamic-decomposition']} for $d=2$ dimensions, $t=2$, and $\Delta=1$. The union of the blue, red, purple, and green region corresponds to the set of generating bases $\mathcal{U}=\left\{10-11,1001,1-101,1011,1-111,1-110\right\}.$We have that $\bigcap_{U\in\mathcal{U}}\mathop{\mathrm{cone}}\nolimits U$ is generated by $v_1=(0,1),v_2=(1,1)$ and that the multiplier $\Psi=2$ suffices to obtain the generating elements $w_1=2v_1=(0,2),w_2=2v_1=(2,2)$ that are in the intersection of integer cones. Furthermore, because $\det W=2$, it suffices to partition the space into lattice translates of $M\mathbb Z^2$ where $M=2\cdot2=4$. The region associated with $\mathcal{U}$ is divided into $4$ subregions based on whether $\lambda_1\ge2$ and/or $\lambda_2\ge2$. For each such subregion, the root lattice point $q$ is marked with a circle. Note that the (degenerate $1$-dimensional) cyan-pink and yellow-orange cones can each be generated by a strictly larger set of bases and are therefore assigned a different set of generating bases $\mathcal{U}$.

Theorems & Definitions (12)

• Conjecture 1: eisenbrand2025parameterizedlinearformulationinteger
• Theorem 2
• Theorem 3
• Theorem 4: DBLP:series/eatcs/Edelsbrunner87DBLP:journals/siamcomp/EdelsbrunnerOS86DBLP:journals/siamcomp/EdelsbrunnerSS93
• Definition 5
• Lemma 6: Lemma 5.3 in DBLP:journals/theoretics/CslovjecsekKLPP25, pottier1991minimal
• Lemma 6
• Theorem 6
• Lemma 7
• Theorem 7