基于邻域的图标记问题
Neighborhood-Aware Graph Labeling Problem
AI总结：
本文研究了基于邻域的图标记问题，证明其NP难性，并提出了在特定条件下多项式时间的近似算法。
摘要：
受带网络干扰的优化或acles启发，我们研究了基于邻域的图标记（NAGL）问题。给定一个图G=(V,E)，一个大小为L的标签集，以及通过评估或acles访问的局部奖励函数f_v，目标是分配标签以最大化Σ_{v∈V}f_v(x_{N[v]})，其中每个项依赖于v的闭邻域。两个顶点出现在某些邻域项中恰好当它们在G中的距离不超过2，因此依赖图是平方图G²，且tw(G²)决定了精确算法和细粒度下界。相应地，我们证明这种依赖是固有的：即使在星图上使用二进制标签，NAGL也是NP难的，并且假设SETH，不存在任何ε>0的(L−ε)^{tw(G²)}⋅n^{O(1)}时间算法。我们通过在G²的树分解上运行精确动态规划来匹配这一结果，时间复杂度为O(n⋅tw(G²)⋅L^{tw(G²)+1})。在近似方面，除非P=NP，对于任何ε>0，即使在一般图上承诺OPT>0，也不存在多项式时间n^{1−ε}近似算法；没有该承诺，不可能有有限的乘法近似比。在非负奖励范围内，我们给出了NAGL在两种情况下的多项式时间近似算法：(i)给定G²的适当q着色，我们获得1/q近似；(ii)在最大度数受限的平面图上，我们开发了一种Baker型多项式时间近似方案（PTAS），当L为常数时，它成为高效的PTAS（EPTAS）。
英文摘要：
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.
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