Motivated by optimization oracles in bandits with network interference, we study the Neighborhood-Aware Graph Labeling (NAGL) problem. Given a graph $G = (V,E)$, a label set of size $L$, and local reward functions $f_v$ accessed via evaluation oracles, the objective is to assign labels to maximize $\sum_{v \in V} f_v(x_{N[v]})$, where each term depends on the closed neighborhood of $v$. Two vertices co-occur in some neighborhood term exactly when their distance in $G$ is at most $2$, so the dependency graph is the squared graph $G^2$ and $\mathrm{tw}(G^2)$ governs exact algorithms and matching fine-grained lower bounds. Accordingly, we show that this dependence is inherent: NAGL is NP-hard even on star graphs with binary labels and, assuming SETH, admits no $(L-\varepsilon)^{\mathrm{tw}(G^2)}\cdot n^{O(1)}$-time algorithm for any $\varepsilon>0$. We match this with an exact dynamic program on a tree decomposition of $G^2$ running in $O\!\left(n\cdot \mathrm{tw}(G^2)\cdot L^{\mathrm{tw}(G^2)+1}\right)$ time. For approximation, unless $\mathsf{P}=\mathsf{NP}$, for every $\varepsilon>0$ there is no polynomial-time $n^{1-\varepsilon}$-approximation on general graphs even under the promise $\mathrm{OPT}>0$; without the promise $\mathrm{OPT}>0$, no finite multiplicative approximation ratio is possible. In the nonnegative-reward regime, we give polynomial-time approximation algorithms for NAGL in two settings: (i) given a proper $q$-coloring of $G^2$, we obtain a $1/q$-approximation; and (ii) on planar graphs of bounded maximum degree, we develop a Baker-type polynomial-time approximation scheme (PTAS), which becomes an efficient PTAS (EPTAS) when $L$ is constant.

翻译：受带网络干扰的赌博机中优化预言机的启发，我们研究邻域感知图标注（NAGL）问题。给定图$G = (V,E)$、大小为$L$的标签集以及通过评估预言机访问的局部奖励函数$f_v$，目标是为顶点分配标签以最大化$\sum_{v \in V} f_v(x_{N[v]})$，其中每项依赖于$v$的闭邻域。两个顶点在某个邻域项中共现当且仅当它们在$G$中的距离不超过$2$，因此依赖图是平方图$G^2$，且$\mathrm{tw}(G^2)$决定了精确算法的复杂度与匹配的精细下界。相应地，我们证明这种依赖性是固有的：即使在二元标签的星形图上，NAGL也是NP难的，并且假设SETH成立，对于任意$\varepsilon>0$，不存在$(L-\varepsilon)^{\mathrm{tw}(G^2)}\cdot n^{O(1)}$时间算法。我们通过基于$G^2$树分解的精确动态规划算法匹配了这一结果，其运行时间为$O\!\left(n\cdot \mathrm{tw}(G^2)\cdot L^{\mathrm{tw}(G^2)+1}\right)$。在近似性方面，除非$\mathsf{P}=\mathsf{NP}$，否则对于任意$\varepsilon>0$，即使在保证$\mathrm{OPT}>0$的条件下，一般图上也不存在多项式时间的$n^{1-\varepsilon}$近似算法；若无$\mathrm{OPT}>0$的保证，则不存在有限乘性近似比。在非负奖励机制下，我们在两种设定中为NAGL提供了多项式时间近似算法：（i）给定$G^2$的一个真$q$染色，我们获得$1/q$近似；（ii）在有界最大度平面图上，我们开发了Baker型多项式时间近似方案（PTAS），当$L$为常数时该方案成为高效PTAS（EPTAS）。