Problem
Research questions and friction points this paper is trying to address.
4-block IPs
FPT
parameterized complexity
4-Block Integer Programming is in FPT
🤖 AI Summary
本文解决了4-块整数规划问题，通过提出一个FPT时间算法，该算法能够处理非线性、可分离凸目标函数，并适用于更广泛的约束矩阵类型。
📝 Abstract
Integer programming is a fundamental and important NP-hard problem. This motivated extensive efforts in studying several tractable subclasses. One of the top unresolved complexity questions is the parameterized complexity of 4-block IPs, a natural class characterized by having a diagonal matrix with small blocks after deleting few rows and columns.
Over the years, significant progress has been made in improving algorithms for 4-block IPs, but the question whether such IPs can be solved in FPT time, parameterized by the block dimensions and largest matrix coefficient, has remained open. This question is repeatedly highlighted, most recently by Koutecký [IPEC 2025] and by Eisenbrand and Rothvoss [SODA 2026].
We resolve this question in the positive by providing an FPT time algorithm that solves general 4-block integer program. Our algorithm can optimize non-linear, separable convex objective functions, and can be extended to broader classes of constraint matrices (such as tree-fold or multi-stage) and allows appending few ``global'' columns to it, and it allows coefficients unbounded by the parameters in those columns. It is known that tractability cannot be extended further in any of those directions. The runtime also nearly matches the known doubly exponential running time lower bound.
The key structural property that we establish is that a function $f\colon\mathbb Z^n\to\mathbb R$ that is integer midpoint convex, i.e., $f(x)\le\tfrac12f(x-p)+\tfrac12f(x+p)$ for all $x,p\in\mathbb Z^n$, can be extended to a convex function on the set $2d\mathbb Z^n\cap L$ if $L$ is a linear subspace of dimension $d$. This closes the gap in a recent work by Ligthart [arXiv 2606.30330, 2026], which allows us to extend the previous algorithm that solves 4-block integer programs with a single global variable to 4-block integer programs that have a parameterized number of global variables.
Innovation
Methods, ideas, or system contributions that make the work stand out.
4-block Integer Programming
FPT Algorithm
Non-linear Convex Objective Functions
Integer Midpoint Convexity
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