A Fixed-Parameter Algorithm for 4-Block Integer Programming

2026-09-20

AI Summary

该研究解决了4-块整数规划问题，通过使用固定参数算法，并基于矩阵最大维度k和绝对值最大的元素Δ作为参数，证实了Eisenbrand和Rothvoss的猜想。

Abstract

We give a fixed parameter tractable (FPT) algorithm with running time $f(k,Δ)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute matrix entry $Δ$. 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 3-block, 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 et al.~(2020) for optimizing discrete convic functions.

Problem

• integer linear programs
• 4-block structure
• parameterized complexity
• maximum block dimension
• absolute matrix entry

Innovation

• Fixed-Parameter Tractable (FPT)
• 4-Block Integer Programming
• Parameterized Complexity