Matchings in Graphs

LP relaxation One way to deal with this is to relax the integrality constraints and allow xe ∈ [0, 1] to get a linear program, which can be solved in polynomial-time. However, this gives rise to fractional matchings. Characteristic vectors of matchings in G can be seen as points in R where m = |E|. The convex hull of all the matchings forms a polytope called the matching polytope M. However, the LP relaxation may give matchings that are outside M. Figure 1 shows some examples. It can be seen that, in examples (1) and (2) in Figure 1, the matching polytope contains all the fractional matchings which form the feasible region of the relaxed LP. However, in Example (3), the maximum value of the relaxed LP is attained at the point ( 2 , 1 2 , 1 2 ), which lies outsideM. We will see next that the matching polytope always contains all the fractional matchings if and only if the graph is bipartite.