We develop a design-based minimax theory for linear causal estimands under interference. We study the minimax risks of three problems: optimizing an estimator under a fixed design, optimizing an estimator and a design from a restricted class, or optimizing both over all designs. In each problem, we study the minimax risk over all estimators and over design-unbiased estimators separately. We allow any correctly specified exposure mapping, including the identity. Our theory represents causal inference under interference as a constrained sampling design on an expanded finite population of $\sum_i d_i$ potential outcomes, where $d_i$ is the number of exposure levels attainable by unit $i$. We describe the constrained sampling design by a weighted sampling hypergraph: its vertices are the potential outcomes, its hyperedges are the maximal sets of potential outcomes that can be jointly sampled under feasible treatment assignments, and its weights are induced by the randomization design. We show that the minimax risks of the fixed-design, restricted-design, and all-design problems depend, respectively, on the weighted hypergraph induced by the design, the collection of weighted hypergraphs attainable within the restricted class, and the unweighted hypergraph. Our Bayes lower bounds compare potential-outcome tables that differ on jointly missing sets of potential outcomes: joint-missingness probabilities govern the lower bounds over all estimators, whereas the corresponding odds govern the lower bounds over unbiased estimators. We illustrate the theory for average treatment effects without interference, saturation designs under partial interference, total, direct, and contagion effects under neighborhood interference, and bipartite dose-response contrasts. The upper and lower bounds match in several settings.

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