Csc5160: Combinatorial Optimization and Approximation Algorithms Topic: Perfect Matching Polytope 12.1 Formulation of General Perfect Matching

In this lecture, the focus is on general perfect matching problem where the goal is to prove that it can be solved in polynomial time by linear programming. Based on the LP formulation for bipartite matching studied in Lecture 10, we add some valid inequalities to establish a new formulation. Then we prove that for general perfect matching, all vertex solutions of the linear program are integral. This shows that the new formulation defines the matching polytope, and the general perfect matching problem can be solved by linear programming. Finally, we will show that this linear program can be solved in polynomial time, by providing a polynomial time separation oracle.