Fixed parameter algorithm solves complex 4-block integer programming puzzles
A Fixed-Parameter Algorithm for 4-Block Integer Programming
Computational ComplexityData Structures and Algorithms
Summary
Integer linear programs help solve many real-world problems by finding solutions that fit certain rules. The authors developed a new algorithm to handle a tricky kind called 4-block integer programs much faster, based on certain problem features. This solves a puzzle that experts had been trying to crack for years. Their method also helps with related problem types, expanding what can be solved efficiently.
What this means in practice
- •For operations research teams: Optimize complex scheduling or resource allocation problems modeled by 4-block integer programs faster by exploiting parameter bounds on block size and matrix entries.
- •For software engineers in logistics: Improve solver performance in routing and supply chain applications that reduce to block-structured integer programs with large but bounded parameter sizes.
A theory result. No direct application yet.
Authors
Klaus Jansen, Felix Ohnesorge, Corinna Wambsganz
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.