We propose an approach for conducting inference for linear unbiased estimators applied to dependent outcomes given constraints on their independence relations, in the form of a dependency graph. We establish the consistency of an oracle variance estimator when a dependency graph is known, along with an associated central limit theorem. We derive an integer linear program for finding an upper bound for the estimated variance when a dependency graph is unknown, but topological or degree-based constraints are available on one such graph. We develop alternative bounds, including a closed-form bound, under an additional homoskedasticity assumption. We establish a basis for Wald-type confidence intervals that are guaranteed to have asymptotically conservative coverage.