The Set Covering Problem selects a minimum-cost subset of sets covering all elements. Given universe U = {1,...,n}, sets S₁,...,Sₘ with costs c₁,...,cₘ, find minimum cost collection covering U. Formulation: minimize Σcⱼxⱼ subject to Σ(j:i∈Sⱼ) xⱼ ≥ 1 for all i, xⱼ ∈ {0,1}. Applications: crew scheduling (cover flight segments), facility location (cover demand points), and emergency service placement. The problem is NP-hard with a greedy ln-approximation algorithm. Solution techniques: branch-and-price (columns are sets), Lagrangian relaxation, and specialized heuristics. Related problems include set packing (disjoint sets) and set partitioning (exact cover), which arise in airline crew scheduling and vehicle routing.