  We propose a general framework based on selectively traversed accumulation
rules (STAR) for interactive multiple testing with generic structural
constraints on the rejection set. It combines accumulation tests from ordered
multiple testing with data-carving ideas from post-selection inference,
allowing for highly flexible adaptation to generic structural information. Our
procedure defines an interactive protocol for gradually pruning a candidate
rejection set, beginning with the set of all hypotheses and shrinking with each
step. By restricting the information at each step via a technique we call
masking, our protocol enables interaction while controlling the false discovery
rate (FDR) in finite samples for any data-adaptive update rule that the analyst
may choose. We suggest update rules for a variety of applications with complex
structural constraints, show that STAR performs well for problems ranging from
convex region detection to FDR control on directed acyclic graphs, and show how
to extend it to regression problems where knockoff statistics are available in
lieu of $p$-values.
