发表机构
Charles University; Eindhoven University of Technology（查理大学; 埃因霍温理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了4-块整数规划可在FPT时间内求解，通过建立整数中点凸函数的凸扩展性质，将算法推广到参数化数量的全局变量，并匹配已知下界。
AI 中文摘要
整数规划是一个基础且重要的NP-hard问题。这促使人们广泛研究其若干可处理的子类。其中一个尚未解决的首要复杂度问题是4-块整数规划的参数化复杂度，这是一个自然类，其特征是在删除少量行和列后具有带小块的对角矩阵。多年来，在改进4-块整数规划的算法方面取得了重大进展，但此类整数规划是否能在以块维度和最大矩阵系数为参数的FPT时间内求解的问题仍然悬而未决。这个问题被反复强调，最近一次是由Koutecký [IPEC 2025]和Eisenbrand与Rothvoss [SODA 2026]提出的。我们通过提供一个求解一般4-块整数规划的FPT时间算法，肯定地解决了这个问题。我们的算法可以优化非线性、可分离的凸目标函数，并且可以扩展到更广泛的约束矩阵类别（如树折叠或多阶段），并允许向其追加少量“全局”列，且允许这些列中的系数不受参数限制。已知在这些方向上的任何进一步扩展都不可能保持可处理性。该运行时间也几乎匹配已知的双指数运行时间下界。我们建立的关键结构性质是：一个函数$f\colon\mathbb Z^n\to\mathbb R$如果是整数中点凸的，即对所有$x,p\in\mathbb Z^n$满足$f(x)\le\tfrac12f(x-p)+\tfrac12f(x+p)$，那么当$L$是维度为$d$的线性子空间时，它可以被扩展为集合$2d\mathbb Z^n\cap L$上的凸函数。这填补了Ligthart [arXiv 2606.30330, 2026]最近工作中的空白，使我们能够将先前求解具有单个全局变量的4-块整数规划的算法扩展到具有参数化数量全局变量的4-块整数规划。
英文摘要
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.
Comments13 pages, 0 figures
