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Help with implementing a randomized approximation algorithm would be appreaciated.

4. Suppose we are given a graph G =(K E), andwe want to label each node u of G with one of k possible labels: 1,2, .. .. lk, where k is a constant. We wish to assign labels to the nodes to maximize the number of edges (u, v) E for which the endpoints, u and v, have different labels. We say that an edge (u, v) is cross if the colors assigned to u and v are different. For example for the following graph, if k 3 the number of cross edges (edges in bold) is 10. of G in such a way that the expected number of cross edges is at least OPT where OPT is the maximum number of edges that can be satisfied (25 marks) Show that the expected nb of satisfied edges is at least lOPT k-1 k-1 4. Suppose we are given a graph G =(K E), andwe want to label each node u of G with one of k possible labels: 1,2, .. .. lk, where k is a constant. We wish to assign labels to the nodes to maximize the number of edges (u, v) E for which the endpoints, u and v, have different labels. We say that an edge (u, v) is cross if the colors assigned to u and v are different. For example for the following graph, if k 3 the number of cross edges (edges in bold) is 10. of G in such a way that the expected number of cross edges is at least OPT where OPT is the maximum number of edges that can be satisfied (25 marks) Show that the expected nb of satisfied edges is at least lOPT k-1 k-1