Welcome to Combinatorics and Optimization

Network meta-analysis (NMA) enables the comparison of multiple medical interventions by combining evidence on their efficacy or safety across clinical trials. Although these models produce rich probabilistic information about how treatments rank, what practitioners often want are simple, interpretable summaries; for example, whether a treatment is likely among the best, or whether one option is likely to outperform another.

The challenge is that, with n treatments, the number of possible questions one can ask about permutations, combinations, or partial orderings of various subsets of treatments grows exponentially. This leads to a large but highly structured combinatorial space, making exhaustive evaluation infeasible.

We develop algorithmic methods to explore this space efficiently and to identify all binary treatment hierarchy statements whose posterior probability exceeds a specified threshold (e.g., 95%). Our approach exploits structure in the ranking space to avoid redundant computations and then prunes conclusions that are logically implied by others, yielding a concise and non-redundant set of results. We illustrate the approach on an NMA of diabetes treatments.