We introduce a general random model of a combinatorial optimization problem with geometric structure that encapsulates both linear programming and integer linear programming. Let $Q$ be a bounded set called the feasible set, $E$ be an arbitrary set called the constraint set, and $A$ be a random linear transform. We define and study the $\ell^q$-margin, $M_q := d_q(AQ, E)$. The margin quantifies the feasibility of finding $y \in AQ$ satisfying the constraint $y \in E$. Our contribution is to establish strong concentration of the margin for any $q \in (2,\infty]$, assuming only that $E$ has permutation symmetry. The case of $q = \infty$ is of particular interest in applications -- specifically to combinatorial ``balancing'' problems -- and is markedly out of the reach of the classical isoperimetric and concentration-of-measure tools that suffice for $q \le 2$. Generality is a key feature of this result: we assume permutation symmetry of the constraint set and nothing else. This allows us to encode many optimization problems in terms of the margin, including random versions of: the closest vector problem, integer linear feasibility, perceptron-type problems, $\ell^q$-combinatorial discrepancy for $2 \le q \le \infty$, and matrix balancing. Concentration of the margin implies a host of new sharp threshold results in these models, and also greatly simplifies and extends some key known results.

翻译：我们提出一个具有几何结构的组合优化问题的通用随机模型，该模型统一涵盖线性规划与整数线性规划。设$Q$为称为可行集的有界集合，$E$为任意称为约束集的集合，$A$为随机线性变换。我们定义并研究$\ell^q$-边际$M_q := d_q(AQ, E)$，该边际量化了在满足约束$y \in E$的条件下寻找$y \in AQ$的可行性。我们的贡献在于建立任意$q \in (2,\infty]$时边际的强集中性，仅需假设$E$具有置换对称性。$q = \infty$情形在应用中具有特殊意义——特别针对组合“平衡”问题——且显著超出足以处理$q \le 2$的经典等周与测度集中工具所能及的范围。该结果的普适性是关键特征：我们仅假设约束集具有置换对称性而无需其他条件。这使得我们能够将众多优化问题（包括以下问题的随机版本：最近向量问题、整数线性可行性、感知器型问题、$2 \le q \le \infty$时的$\ell^q$-组合差异度量、矩阵平衡）用边际进行编码。边际的集中性在这些模型中导出大量新的尖锐阈值结果，同时极大简化并推广了一些关键已知结论。