Given a linear program (LP ) with m constraints and n lower and upper bounded variables, any solution \(x^0\) to LP can be represented as a nonnegative combination of at most \(m + n\) so-called weighted paths and weighted cycles, among which at most n weighted cycles. This fundamental decomposition theorem leads us to derive, on the residual problem LP (\(x^0\) ), two alternative optimality conditions for linear programming, and eventually, a class of primal algorithms that rely on an Augmenting Weighted Cycle Theorem.