A short course on matching

Linear programming is a very powerful technique, applicable not only in optimization, but in combinatorial theory. For example, we will see that König’s theorem (Lecture 1,Theorem 2) is an instance of the principle of (integral) linear programming duality. In this section, we remind ourselves of the basic relevant concepts. Given a collection of vectors v1, v2, . . . .vm ∈ R their linear combination is λ1v1 + λ2v2 + . . . + λmvm ∈ R, where λ1, . . . , λm ∈ R. The linear combination is affine if λ1 + . . . + λm = 1, it is convex if additionally all λi ≥ 0. The convex hull of a family of vectors is the set of all convex combinations of vectors in this family. The convex hull of a finite family of vectors is called a polytope. A classical theorem of Weil states that every polytope is an intersection of a finite collection of half spaces. Thus every polytope can be characterized either by a (minimal) set of vectors whose convex combinations generate this polytope or by a family of half spaces. The dimension of a polytope is the dimension