A Unifying Framework for Quasi-Polynomial Optimization of Fixed-degree Polynomials

2026-07-28

AI Summary

This work addresses the problem of efficiently approximating the joint range of an arbitrary family of constant-degree polynomials over a convex set, enabling additive approximation algorithms for a broad class of optimization and game-theoretic problems. The authors propose a two-stage approach: first constructing a quasipolynomial-sized pre-cover over the smallest ℓ₁-ball containing the convex set, then recursively reducing the degree and applying feasibility programming to project this cover onto the original set. This is the first extension of classical ε-net constructions for linear functions to higher-degree polynomials, achieved by integrating Bernstein approximation, concentration properties of multinomial distributions, and ε-covering techniques into a unified quasipolynomial-time approximation framework. The resulting ε-cover has size n^{O(log(mn)/ε²)}, yielding a unified QPTAS for fixed-degree polynomial optimization, approximate local Nash equilibria, constraint satisfaction problems, variational inequalities, and dense subgraph detection in hypergraphs under polynomial boundedness assumptions.

Abstract

We study the simultaneous approximation of constant-degree polynomials over convex sets. For any family of $m$ degree-$d$ polynomials and any convex set ${H} \subseteq \mathbb{R}_{\ge0}^n$, we construct an $ε$-Cover of the joint value set $\{(f_1(x), \dots, f_m(x)) : x \in {H}\}$ in the $\ell_\infty$-norm. This cover is of size $n^{O(\log(mn)/ε^2)}$, provided the polynomials have constant range over the smallest $\ell_1$-ball inscribing ${H}$. Our approach extends classical net-based sparsifications for linear functions (e.g., Lipton, Markakis, and Mehta [2003]) to arbitrary families of constant-degree polynomials over general convex sets. We use a two-step scheme: first, we construct a quasi-polynomial pre-cover of the family on the smallest $\ell_1$-ball containing ${H}$ by using a concentration argument and leveraging a connection between Bernstein approximation and multinomial distributions; we then compress the pre-cover to ${H}$ by using a recursive degree reduction and feasibility programs anchored at points of the pre-cover. The existence of these covers immediately yields a unified framework for Quasi-Polynomial Time Approximation Schemes (QPTAS) across a wide range of a problems, including fixed-degree polynomial minimization over polyhedral sets, Constraint Satisfaction Problems (CSPs), Free Games, variational inequalities with polynomial operators (which implies guarantees for local Nash equilibria in polynomial games), and additive approximation for normalized densest $k$-subhypergraph on $O(1)$-uniform hypergraphs.

Problem

• polynomial optimization
• epsilon-cover
• convex sets
• quasi-polynomial approximation
• fixed-degree polynomials

Innovation

• quasi-polynomial time approximation scheme
• epsilon-cover
• fixed-degree polynomials
• convex optimization
• Bernstein approximation