Offline Reachability in Directed Acyclic Graphs under Vertex Failures

We study an offline reachability problem in a directed acyclic graph: given a common source and failure sets paired with target sets, does any query retain a reachable target? This is linearly equivalent to finding a path to a labelled terminal that avoids every vertex forbidding its label. For queries with at most two failures, let V, E, q_b>=1 and M count vertices, arcs, queries and target incidences, and put S=V+E+q_b+M. For every integer tau>=1, our threshold algorithm uses Õ(S+E tau+(V+E)q_b/tau) time and Õ(S+V tau) working words, counting discovery, edge scans and target aggregation. Here Õ suppresses polylogarithmic factors. Setting tau=ceil(sqrt(q_b)) gives Õ(V^(3/2)) time when E, q_b and M are O(V). Exact sharing of identical stored sets preserves the time bound, but can still require superlinear space. Further algorithms give near-linear time if the DAG has polylogarithmic depth or a supplied partition into polylogarithmically many sequences in which each vertex reaches the next. For unrestricted multiplicity, clique and directed-cycle reductions give conditional reachability bounds at specified label-count regimes, distinguishing combinatorial from general word random-access machine algorithms. We also study falsification of guarded Boolean expressions: the listed Hitting Set Conjecture gives a conditional lower bound for falsifying at least one of several shared outputs. Near-linear total time for arbitrary two-failure batches, and near-linear space at the square-root-threshold time bound, remain open.