Zero-One Laws for Random Feasibility Problems

TL;DR

This work introduces a universal random feasibility framework built around the ℓ^q-margin $M_q(A)=\min_{\sigma\in Q} d_q(A\sigma,E)$, where $A$ has iid Gaussian entries and $Q,E$ encode a wide class of combinatorial problems. Using the envelope theorem to differentiate the margin and Talagrand’s Gaussian $L^1$-$L^2$ inequality to bound its variance, the authors obtain strong concentration results for all $q\in[2,\infty]$ under permutation symmetry of $E$, with a nearly optimal rate for $q=\infty$ and a natural extension to block-symmetric $E$. These variance bounds translate into sharp threshold phenomena for feasibility across diverse models, including random integer programming, combinatorial discrepancy, perceptron-type problems, and matrix balancing, often yielding simpler or sharper results than prior approaches. The paper thus unifies and extends a broad swath of high-dimensional random optimization problems, while also outlining open directions such as relaxing symmetry assumptions and analyzing non-Gaussian disorder in the matrix $A$.

Abstract

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.

Theorems & Definitions (25)

• Definition 1: Random Feasibility Problem
• Definition 2
• Theorem 1: Main result: concentration of the margin
• Theorem 2: Block symmetry suffices
• Remark 1
• Theorem 3: Sharp threshold for $\ell^q$ discrepancy
• proof
• Theorem 4: Sharp threshold for the symmetric spherical perceptron
• Theorem 5: Sharp threshold for the capacity of the generalized perceptron nakajima-sunxu
• Theorem 6: Sharp threshold for the margin of the generalized perceptron