arXiv:2609.23711 · v2

A Fixed-Parameter Algorithm for 4-Block Integer Programming

首次提交 2026-09-20 23:41

版本更新 2026-11-03 15:12

雷达观测 2026-11-04 23:40

摘要

We give a fixed-parameter tractable (FPT) algorithm with running time $f(k,\barΔ)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute entry $\barΔ$ in the local-column blocks. This result resolves a long-standing open question in parameterized complexity, and answers a conjecture by Eisenbrand and Rothvoss (2026) in the positive. This result has several implications for other block-structured integer programming models, including mixed fracture number and special cases of 4-block programs with large entries outside of the diagonal. Important tools for this algorithm are structural properties of generalized $n$-fold integer programs shown by Ligthart~(2026) and an algorithm by Veselov, Gribanov, Zolotykh and Chirkov~(2020) for optimizing discrete convic functions. Independently and shortly after the first version of our paper appeared, Koutecký, Lassota, and Ligthart (2026) obtained an FPT algorithm for 4-block integer programming.