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 introduce and study the $\ell^q$-\textit{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：我们提出了一个具有几何结构的组合优化问题的通用随机模型，该模型同时涵盖了线性规划和整数线性规划。设$Q$为称为可行集的有界集，$E$为称为约束集的任意集，$A$为随机线性变换。我们引入并研究了$\ell^q$-边际$M_q := d_q(AQ, E)$。该边际量化了寻找满足约束条件$y \in E$的$y \in AQ$的可行性。我们的贡献在于，在仅假设$E$具有置换对称性的条件下，建立了对任意$q \in (2,\infty]$的边际强集中性质。$q = \infty$的情况在应用中——特别是组合"平衡"问题中——具有特殊意义，并且明显超出了足以处理$q \le 2$情形的经典等周和测度集中工具所能达到的范围。普适性是这一结果的关键特征：我们仅假设约束集具有置换对称性，再无其他条件。这使得我们可以将许多优化问题用边际来编码，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，以及矩阵平衡。边际的集中性质在这些模型中导出大量新的尖锐阈值结果，同时也极大地简化和扩展了若干已知的关键结果。