Question: 3. (20 pts) Given an undirected graph G = (V, E), a matching M in G is a set of pairwise non- adjacent edges; that

3. (20 pts) Given an undirected graph G = (V, E),

3. (20 pts) Given an undirected graph G = (V, E), a matching M in G is a set of pairwise non- adjacent edges; that is, no two edges share a common vertex. The maximum matching problem is a matching that contains the largest possible number of edges. By defining a binary decision variable Zij {0,1} to indicate whether edge (i, j) is chosen into the matching or not, this problem can be formulated as the linear program by relaxing Xij: 2 max Xij (i,j)EE s.t. I Bij 0, Hi,j) E E Find the dual of this problem (which should be the LP relaxation of vertex cover problem). = Hint: you may use the graph G = (V, E) with V = {1,2,3,4}, E = {(1,2), (1,3), (2,3), (2,4),(3,4)} as : ) an example to find the dual problem for matching problem, and then use the idea to find the general dual problem. (For undirected graph, C12 and 221 are the same variable)

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