Consider the following computational problem defined on a path spanning n nodes

Question

Consider the following computational problem defined on a path spanning n nodes labeled u_1, u_2,...,u_n Each pair of vertices (u_i, u_j) has associated a profit p(i, j) and a bandwidth requirement b(i, j) for communicating with one another (when a pair (u_i, u_j) communicates it used b(I, j) bandwidth in each edge between u_i and u_j). Each edge in the path has a bandwidth capacity of B. We are interested in the problem of finding a maximum profit subset of pairs of vertices such that all pairs can communicate simultaneously without violating the edge bandwidth capacities. More formally, we want to select a set with maximum profit such that for every edge (u_k, u_k+1) in the backbone we have Prove that this problem is NP-hard.

Explanation / Answer

Solution : To prove that the problem is NP-Hard, if we consider the different number of spanning trees which can flow the maximum profit then we can say the problem is NP- Hard.

Just consider the two spanning tree :

1) if i=0, j=1 ; (ui,uj)

2) if i=1, j=2 ; (ui,uj)

After putting values of i and j and getting the two spanning tree, calculate the cost and gets the maximum profit spanning tree. By this we can have that the problem is a NP-Hard problem.

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