Linear Programming

Introduction to Linear Programming Linear programming is a very important class of problems, both algorithmically and combinatorially. Linear programming has many applications. From an algorithmic point-of-view, the simplex was proposed in the forties (soon after the war, and was motivated by military applications) and, although it has performed very well in practice, is known to run in exponential time in the worst-case. On the other hand, since the early seventies when the classes P and NP were defined, it was observed that linear programming is in NP" co-NP although no polynomial-time algorithm was known at that time. The first polynomial-time algorithm, the ellipsoid algorithm, was only discovered at the end of the seventies. Karmarkar's algorithm in the mid-eighties lead to very active research in the area of interior-point methods for linear programming. We shall present one of the numerous variations of interior-point methods in class. From a combinator