Network Design Foundation Homework 2 - Solution
نویسنده
چکیده
1. • Let T = (V, E) be a tree and let D =< d1, . . . , dk > denote the set of demands. We reduce the prize-collecting steiner forest problem on trees to the minimum weighted vertex cover problem. We make a bipartite graph H = (V ′, E ′) where for each edge e ∈ E we put a vertex pe in V ′ and also for every di ∈ D we put a vertex qi in V ′. We put an edge between pe and qi iff the edge e is on the unique path between the endpoints of demand di in T . The weight of pe would be the length of e and the weight of qi would be the prize of di. Recall that a vertex cover is a subset S of vertices which every edge has at least an endpoint in S. One can easily show that the cost of the minimum vertex cover in H is equal to the cost of the minimum steiner forest in T . We note that for any vertex qi, we have to include qi in our vertex cover unless we have chosen all the pe s adjacent to qi which essentially means that we have chosen all the edges in the path between the endpoints of di. Vertex cover can be modeled as a set cover problem where the frequency of each member is two, and thus it has a 2-approximation algorithm (consider the edges as elements and the vertices as sets). However since H here is a bipartite graph there is a polynomial time exact algorithm based on primal-dual ideas. • For general graphs, we simply embed the graph into a collection of trees, paying at most a factor of O(logn) on distances and thus on the approximation ratio. Then we solve the problem on trees with the 2-approximation algorithm (or the exact algorithm) described above.
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