Decomposition theorems for linear programs.

Given a linear program  (LP) with m constraints and n lower and upper bounded variables, any solution x0 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(x0), two alternative optimality conditions for linear programming, and eventually, a class of primal algorithms that rely on an Augmenting Weighted Cycle Theorem.