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原文摘要

Integer programming is a fundamental and important NP-hard problem. This motivated extensive efforts in studying several tractable subclasses. One of the most important, 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. 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 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 incorporate any parameterized number of global variables. As an example, we apply our algorithm to solve an open problem posed by Boehmer and Koana [ACM Trans. Algorithms, 2024] in Max-Min Fair Matching.