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

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.

💡 Research Summary

The paper tackles the long‑standing challenge of efficiently solving 4‑block integer linear programs (ILPs), a class of structured ILPs that arise in many combinatorial optimization problems. A 4‑block ILP consists of a global block of variables (x_{0}\in\mathbb Z^{s}) and (n) local blocks (x_{i}\in\mathbb Z^{t}) (for (i=1,\dots,n)). The constraint matrix has the form

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