TCARD: Nearly Balanced Two-Level Designs with Treatment Cardinality Constraints with an Application to LLM Prompt Engineering
Modern experimental designs often face the so-called treatment cardinality constraint, which is the constraint on the number of included factors in each treatment. Experiments with such constraints are commonly encountered in engineering simulation, AI system tuning, and large-scale system verification. This calls for the development of adequate designs to enable statistical efficiency for modeling and analysis within feasible constraints. In this work, we study two-level designs under this $k$-treatment cardinality constraint (TCARD), where the design matrix $\mathbf{X} \in \{0,1\}^{n \times p}$ has constant row sums equal to $k$. Although TCARDs are closely related to balanced incomplete block designs (BIBDs), exact BIBD structure is unavailable for many practical $(n,p,k)$ combinations. This leads to the notion of nearly balanced TCARDs, which we prove minimize the first two components of the generalized word-length pattern. We also show that good projection behavior in this setting is governed by two count-based regularities: balanced factor replications and uniform pairwise concurrences. Motivated by this characterization, we then propose the Balanced Concurrence Deviation ($Φ_{\mathrm{BCD}}$), a model-free objective that jointly penalizes replication imbalance and concurrence dispersion. We further show that this criterion is closely connected to classical optimality principles, including $(M,S)$-optimality, centered $\mathrm{UE}(s^2)$ criterion, and Bayesian $D$-optimality. To construct designs minimizing $Φ_{\mathrm{BCD}}$, we develop a coordinate-exchange (CE) algorithm with efficient incremental updates, together with a simulation-based procedure for calibrating the criterion weights to the intended downstream task. Numerical experiments confirm that the proposed method compares favorably with existing alternatives across a range of problem sizes and constraint strengths.