Neighborhood-Aware Graph Labeling Problem

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.