Data-flow analysis computes its solutions over the paths in a control-flow graph. These paths---whether feasible or infeasible, heavily or rarely executed---contribute equally to a solution. However, programs execute only a small fraction of their potential paths and, moreover, programs' execution time and cost is concentrated in a far smaller subset of hot paths.This paper describes a new approach to analyzing and optimizing programs, which improves the precision of data flow analysis along hot paths. Our technique identifies and duplicates hot paths, creating a hot path graph in which these paths are isolated. After flow analysis, the graph is reduced to eliminate unnecessary duplicates of unprofitable paths. In experiments on SPEC95 benchmarks, path qualification identified 2--112 times more non-local constants (weighted dynamically) than the Wegman-Zadek conditional constant algorithm, which translated into 1--7% more dynamic instructions with constant results.