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A. (50 PTS.) We are given a directed graph G with n vertices and m edges (m 〉 n), where each vertex v has a height h(v). The cost of traversing an edge (u, v) is c(u, v) – Ih(v) -h(u)l The cost of a walk in G is the sum of the costs of edges in the walk. Prove that finding the minimum cost walk that visits all the vertices om G is NP-HARD. (In a walk, vertices and edges may be repeated, and the start and end vertices may be different .) B. (50 PTS.) We are given a directed graph G with n vertices and m edges (m-n), where each edge e has a set of colors C(e) S [1,. ky. Prove that deciding whether there exists a walk that uses all k colors (i.e. the union of the sets of colors of the edges of walk covers all colors.) is NP-HARD. (Hint: Reduce from Set Cover.) Show transcribed image text A. (50 PTS.) We are given a directed graph G with n vertices and m edges (m 〉 n), where each vertex v has a height h(v). The cost of traversing an edge (u, v) is c(u, v) – Ih(v) -h(u)l The cost of a walk in G is the sum of the costs of edges in the walk. Prove that finding the minimum cost walk that visits all the vertices om G is NP-HARD. (In a walk, vertices and edges may be repeated, and the start and end vertices may be different .) B. (50 PTS.) We are given a directed graph G with n vertices and m edges (m-n), where each edge e has a set of colors C(e) S [1,. ky. Prove that deciding whether there exists a walk that uses all k colors (i.e. the union of the sets of colors of the edges of walk covers all colors.) is NP-HARD. (Hint: Reduce from Set Cover.)
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