AI 中文总结
研究凸集上固定次数多项式同时逼近，通过两步方案构造\(\epsilon\)覆盖，大小为\(n^{O(\log(mn)/\epsilon^2)}\)，为多类问题提供拟多项式时间近似方案统一框架。
AI 中文摘要
我们研究凸集上固定次数多项式的同时逼近问题。对于任意\(m\)个\(d\)次多项式族和任意凸集\(H\subseteq\mathbb{R}_{\geq0}^n\)，我们在\(\ell_{\infty}\)范数下构造联合值集\(\{(f_1(x),\dots,f_m(x)):x\in H\}\)的\(\epsilon\)覆盖。该覆盖大小为\(n^{O(\log(mn)/\epsilon^2)}\)，前提是多项式在包含\(H\)的最小\(\ell_1\)球上具有常数范围。我们的方法将线性函数的经典基于网的稀疏化扩展到一般凸集上任意固定次数多项式族。我们采用两步方案：首先，通过集中论证并利用伯恩斯坦逼近与多项分布之间的联系，在包含\(H\)的最小\(\ell_1\)球上构造该族的拟多项式预覆盖；然后通过递归降次和以预覆盖点为锚点的可行性程序将预覆盖压缩到\(H\)上。这些覆盖的存在立即为广泛问题产生了拟多项式时间近似方案（QPTAS）的统一框架，包括多面体集上的固定次数多项式最小化、约束满足问题（CSP）、自由博弈、多项式算子的变分不等式（这意味着多项式博弈中局部纳什均衡的保证）以及\(O(1)\)均匀超图上归一化最密集\(k\)子超图的加法逼近。
英文摘要
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.
