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，我们研究了邻域感知图标注（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。当两个顶点在图 $G$ 中的距离不超过 $2$ 时，它们会在某个邻域项中同时出现，因此依赖图是平方图 $G^2$，且 $\mathrm{tw}(G^2)$ 决定了精确算法与匹配的精细粒度下界。相应地，我们证明了这种依赖性是固有的：即使在具有二元标签的星形图上，NAGL 也是 NP 难问题，并且在假设 SETH 的前提下，对于任意 $\varepsilon>0$，不存在 $(L-\varepsilon)^{\mathrm{tw}(G^2)}\cdot n^{O(1)}$ 时间算法。我们通过一个在 $G^2$ 的树分解上运行的精确动态规划算法匹配了这一结果，其时间复杂度为 $O\!\left(n\cdot \mathrm{tw}(G^2)\cdot L^{\mathrm{tw}(G^2)+1}\right)$。在近似算法方面，除非 $\mathsf{P}=\mathsf{NP}$，否则对于任意 $\varepsilon>0$，即使在满足 $\mathrm{OPT}>0$ 承诺的一般图上，也不存在多项式时间的 $n^{1-\varepsilon}$ 近似算法；若不满足 $\mathrm{OPT}>0$ 承诺，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 给出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）在有界最大度的平面图上，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）。