Learning to Cover Locally: Graph Neural Combinatorial Optimization under a Hard Information Horizon

Neural combinatorial optimization typically assumes a centralized solver that reads the whole instance. We study the opposite: combinatorial optimization under a hard information horizon, where every node commits to its share of a global solution seeing only its k-hop neighborhood, and those commitments must compose into a globally feasible solution. We formalize this as local set cover and instantiate it on weighted multipoint relay (MPR) selection, the NP-hard 2-hop covering problem of the Optimized Link State Routing Protocol version 2 (OLSRv2) routing protocol (RFC 7181), whose horizon is imposed by the protocol, not chosen by the modeler. We prove two results. Any deterministic selector whose horizon is one hop short must either fail coverage or land a factor Δ from optimal, and an L-layer graph neural network (GNN) read out at the deciding node is exactly an L-hop selector, so capacity cannot buy back radius. Conversely, at the horizon a of depth O(Δ) reproduces the RFC 7181 covering greedy, and at width O(cΔ) its metric-aware weighted analogue, inheriting the (1+ Δ 2) -approximation in both cases. Empirically, a 3-layer with a coverage-completing decoder, behavior-cloned from the CP-SAT optimum, reaches cost/opt=1.030 plus or minus0.001 against greedy's 1.138, closing 79.1% of the gap at 100% coverage. Restricting the same learner to one hop, on identical instances with the same decoder and demonstrations, collapses it to 1.344, far worse than greedy. Two transfer checks target real-world networks. OLSRv2's unmodified selection code matches our cardinality greedy on 200/200 unit-cost instances, and on 40, 308 instances of real battalion mobility the frozen model closes 48% of the gap at full coverage. The information horizon, not the model capacity, is the most significant variable.