LP Relaxation transforms an integer program into a linear program by relaxing integrality restrictions, allowing variables to take continuous values. For binary variable x ∈ {0,1}, relaxation gives 0 ≤ x ≤ 1. The LP relaxation provides: (1) lower bound on optimal integer solution value (for minimization); (2) indicator of problem difficulty (large gap suggests hard problem); (3) basis for branch-and-bound algorithms. If LP relaxation solution is integer, it's optimal for the original IP. Strong formulations have tight LP relaxations (small gap). Techniques to strengthen: adding valid inequalities, reformulation, and extended formulations with additional variables. LP relaxation quality fundamentally determines integer programming solver performance.