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：我们提出了一种具有几何结构的组合优化问题的一般随机模型，该模型同时囊括了线性规划和整数线性规划。设$Q$为有界集（称为可行集），$E$为任意集（称为约束集），$A$为随机线性变换。我们定义并研究$\ell^q$-间隔$M_q := d_q(AQ, E)$。该间隔量化了寻找满足约束$y \in E$的$y \in AQ$的可行性。我们的贡献在于：仅假设$E$具有置换对称性，即可对任意$q \in (2,\infty]$建立间隔的强集中性。$q = \infty$的情形在应用中尤其重要——特别针对组合“平衡”问题——且完全超出了经典等周不等式与测度集中工具的处理范围（这些工具对$q \le 2$的情形已足够）。该结果的普适性是其关键特征：我们仅假设约束集具有置换对称性，无需其他条件。这使得我们能够用间隔来编码许多优化问题，包括以下问题的随机版本：最近向量问题、整数线性可行性、感知器类问题、$2 \le q \le \infty$的$\ell^q$-组合差异问题，以及矩阵平衡。间隔的集中性意味着这些模型中一系列新的尖锐阈值结果，同时也极大地简化和扩展了某些已知的关键结果。