Neighborhood-Aware Graph Labeling Problem

Abstract: 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 $G<sup>2$ and $\mathrm{tw}(G<sup>2)$ 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 $$(L-\varepsilon)^{{\mathrm{tw}(G^2)}\cdot} n<sup>{O(1)}$$-time algorithm for any $\varepsilon&gt;0$. We match this with an exact dynamic program on a tree decomposition of $G<sup>2$ running in $$O!\left(n\cdot \mathrm{tw}(G^2)\cdot L<sup>{\mathrm{tw}(G<sup>2)+1}\right)$$ time. For approximation, unless $\mathsf{P}=\mathsf{NP}$, for every $\varepsilon&gt;0$ there is no polynomial-time $n<sup>{1-\varepsilon}$-approximation on general graphs even under the promise $\mathrm{OPT}&gt;0$; without the promise $\mathrm{OPT}&gt;0$, 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 $G<sup>2$, 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.

• The paper characterizes NAGL as optimization over the squared graph G², proving NP-hardness on binary-labeled stars and a SETH-tight lower bound of (L−ε)^tw(G²)·n^{O(1)}.
• The paper introduces Clique-Focused Dynamic Programming, which finds exact solutions in O(n·t·L^{t+1}) time for treewidth t and matches the optimal exponential dependence on label count.
• For nonnegative rewards, the paper provides 1/q coloring approximations and a planar bounded-degree PTAS, while showing general multiplicative approximation is impossible without additional assumptions.

The paper studies the Neighborhood-Aware Graph Labeling (NAGL) problem, an abstract combinatorial optimization formulation of the optimization oracle that arises in multi-armed bandits with network interference (2602.08098). Given an undirected graph $G=(V,E)$, a label set $L$ with $L \coloneqq |L|$, and local reward functions $f_v : L^{|N[v]|} \to \mathbb{R}$ accessible via polynomial-time evaluation oracles, NAGL asks for a labeling maximizing $F(x)=\sum_{v\in V} f_v(x_{N[v]})$, where each term depends on the closed neighborhood of a vertex. Because two vertices co-occur in some local term exactly when their distance in $G$ is at most two, the dependency structure is the squared graph $G^2$; equivalently, NAGL is a weighted constraint satisfaction problem whose primal graph is $G^2$. The paper delivers a tight characterization of both exact and approximate solvability of this oracle, which prior work on interference-aware learning had assumed away as computationally free.

Hardness: NP-hardness on stars and SETH-tight lower bounds

The central hardness construction reduces $k$-SAT to NAGL on a star graph. The reduction packs $N$ Boolean variables into base-$L$0 digits distributed over $L$1 leaves; the center's reward function decodes the leaf labels into a truth assignment and evaluates $L$2. All rewards are in $L$3, $L$4 iff $L$5 is satisfiable, and since any two leaves are at distance two, $L$6 is the clique $L$7 with $L$8. This yields two strong statements. First, NAGL is NP-hard even on star graphs with binary labels ($L$9). Second, assuming SETH, no algorithm runs in time $L \coloneqq |L|$0 for any $L \coloneqq |L|$1, for every fixed $L \coloneqq |L|$2.

Two caveats deserve emphasis. The reduction is polynomial-time only under the evaluation-oracle input model: if $L \coloneqq |L|$3 were given explicitly as a table, the input size would be $L \coloneqq |L|$4, itself exponential in $L \coloneqq |L|$5. Also, bounding $L \coloneqq |L|$6 alone does not control complexity—a star has $L \coloneqq |L|$7 but $L \coloneqq |L|$8—and neither does bounding $L \coloneqq |L|$9 alone, since bounded-degree expanders have $f_v : L^{|N[v]|} \to \mathbb{R}$0. Tractability requires both quantities small; the useful upper bound $f_v : L^{|N[v]|} \to \mathbb{R}$1 follows from a bag-expansion argument.

Inapproximability

With rewards allowed to be negative—or even with $f_v : L^{|N[v]|} \to \mathbb{R}$2 rewards but without a promise on the optimum—multiplicative approximation collapses entirely. A gap-at-zero argument shows that unless $f_v : L^{|N[v]|} \to \mathbb{R}$3, no polynomial-time algorithm achieves any finite multiplicative approximation ratio, even on stars with binary labels, because distinguishing $f_v : L^{|N[v]|} \to \mathbb{R}$4 from $f_v : L^{|N[v]|} \to \mathbb{R}$5 decides SAT. Under the promise $f_v : L^{|N[v]|} \to \mathbb{R}$6, a clean reduction from Maximum Independent Set (labelings encode independent sets so that $f_v : L^{|N[v]|} \to \mathbb{R}$7) shows that for every $f_v : L^{|N[v]|} \to \mathbb{R}$8 there is no polynomial-time $f_v : L^{|N[v]|} \to \mathbb{R}$9-approximation on general graphs, again with $F(x)=\sum_{v\in V} f_v(x_{N[v]})$0. These results imply that meaningful approximation necessarily exploits structure beyond general graphs—either nonnegative rewards with sparsity, or planarity.

Exact algorithm: Clique-Focused Dynamic Programming

On the algorithmic side, the paper introduces Clique-Focused Dynamic Programming (CFDP), which runs over a nice tree decomposition of $F(x)=\sum_{v\in V} f_v(x_{N[v]})$1 of width $F(x)=\sum_{v\in V} f_v(x_{N[v]})$2. The key structural fact is that each closed neighborhood $F(x)=\sum_{v\in V} f_v(x_{N[v]})$3 is a clique in $F(x)=\sum_{v\in V} f_v(x_{N[v]})$4 and hence is contained in some bag; assigning each reward term $F(x)=\sum_{v\in V} f_v(x_{N[v]})$5 to such a bag decomposes $F(x)=\sum_{v\in V} f_v(x_{N[v]})$6 as a sum of per-bag functions $F(x)=\sum_{v\in V} f_v(x_{N[v]})$7, after which standard bottom-up recurrences over introduce, forget, and join nodes apply. CFDP computes an optimal labeling in $F(x)=\sum_{v\in V} f_v(x_{N[v]})$8 time and $F(x)=\sum_{v\in V} f_v(x_{N[v]})$9 space ($G$0 working space if only the value is needed). Combined with the SETH lower bound, this establishes that the base $G$1 in the exponential dependence on $G$2 is optimal up to polynomial factors—a matching upper and lower bound in the fine-grained sense.

Coloring-based approximation for nonnegative rewards

Assuming all $G$3, the paper gives a simple color-class maximization scheme. Within a color class of a proper coloring of $G$4, closed neighborhoods are pairwise disjoint (vertices at distance at least three), so per-vertex neighborhood maximizers can be combined into one consistent labeling without conflicts. Given a proper $G$5-coloring of $G$6, this yields a $G$7-approximation using $G$8 oracle evaluations, i.e., $G$9 time. Without a supplied coloring, a greedy coloring gives a $G^2$0-approximation, hence a constant-factor guarantee when $G^2$1 is bounded via $G^2$2. These guarantees are nonvacuous only under $G^2$3, an assumption the paper states explicitly.

Baker-type PTAS on planar bounded-degree graphs

For planar graphs with bounded maximum degree and nonnegative rewards, the paper develops a Baker-type shifting scheme. Removing one BFS layer out of every $G^2$4, exactly solving NAGL on each remaining component with CFDP, and taking the best over all $G^2$5 offsets yields a $G^2$6-approximation with $G^2$7. The loss factor of $G^2$8 comes from radius-one dependence: a vertex is unsafe under an offset only if its own layer or an adjacent layer is removed. Components of the punctured graph have $G^2$9 by diameter-treewidth bounds for minor-closed families, giving $G^2$0 and overall running time $G^2$1; for constant $G^2$2 this is an EPTAS. An extension remark covers radius-$G^2$3 dependencies, where the shifting loss becomes $G^2$4.

Budgeted submodular variant and experiments

An appendix treats binary labels with budgeted monotone submodular objectives: local submodularity of each $G^2$5 on $G^2$6 lifts to global submodularity of $G^2$7, so Standard Greedy attains the Nemhauser–Wolsey–Fisher $G^2$8 guarantee.

Experiments support the theory. On Minnesota road subgraphs with growing $G^2$9, CFDP outperforms a Gurobi-based ILP throughout (15 seconds versus 16 minutes at $k$0), though both eventually hit a 10-hour limit (CFDP at $k$1, ILP at $k$2). On $k$3 ladder graphs, where $k$4 is constant, CFDP scales to $k$5 in roughly 20 seconds while the ILP times out already near $k$6, confirming the predicted dependence on treewidth rather than on $k$7 alone. In the budgeted submodular setting on US road subgraphs up to $k$8, Greedy runs in about 21 seconds (versus roughly 30 minutes for ILP) and empirically achieves approximation ratios around $k$9, above the worst-case $N$0.

Limitations and open problems

The paper's guarantees depend on explicit assumptions at several points: hardness and exactness rely on the evaluation-oracle model for reward functions; approximation results require nonnegativity (and $N$1 for multiplicative ratios to be meaningful); the PTAS requires planarity and bounded degree; and CFDP assumes a width-$N$2 tree decomposition of $N$3 is available, with an additional $N$4 overhead for computing one. Notably, the approximation guarantees are driven purely by graph structure; the paper leaves open whether structure in the reward functions themselves—such as exposure mappings or other low-complexity neighborhood summaries arising in the interference literature—admits polynomial-time approximation algorithms, possibly even on general graphs. It also leaves open extending CFDP to budgeted set-function objectives without inflating the state space by a factor of $N$5.

Conclusion

NAGL isolates the computational core of optimization oracles used in bandits with network interference and characterizes it tightly: NP-hard on stars with binary labels, SETH-tight at base $N$6 parameterized by $N$7, essentially inapproximable on general graphs, yet admitting a $N$8-coloring approximation and a PTAS/EPTAS on planar bounded-degree graphs with nonnegative rewards. The results indicate that the practicality of interference-aware learning algorithms hinges on structural parameters of the squared graph, not of the original network.