By Witold Bednorz
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Additional info for Advances in Greedy Algorithms
Garey and D. S. Johnson. “Computers and Intractability: A Guide to the Theory of NP-Completeness”. H. Freeman Publishing Company, 1979. 38 Advances in Greedy Algorithms  Y. Bejerano abd R. Rastogi, “Robust monitoring of link delays and faults in IP networks”. IEEE/ACM Trans. on Networking, Vol. 14, No. 5, pp 1092-1103, 2006.  U. Feige, “A threshold of ln n for approximating set-cover”, Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 314-318, 1996.  B. M. Waxman.
The variables in P0 are visited in a random order. If a variable has not been matched yet, then we randomly select another unmatched variable, and a multivariable consisting of these two variables is created. Unmatchable variables are simply copied to the coarser level. The new multivariables are used to define a new and smaller problem. This coarsening process is recursively carried out until the size of the problem reaches some desired threshold. Initial solution: An initial assignment Am of Pm is easily computed using a random assignment algorithm, which works by randomly assigning to each multivariable of the coarsest problem Pm the value of true or false.
Now we will show that the given VC instance, graph G(V,E), has a solution, S of size k if and only if the PM instance, graph G ( V , E ) has a solution, S of size k + 2. In this proof we assume without lose of generality that the routing tree (RT) of every node is its shortest path tree (SPT). First, let considered the auxiliary structure defined by the nodes in R. The edge (r2, r3) is covered only by the SPTs and . Therefore, one of these nodes must be 4 5 included in S . Similarly, one of the nodes r or r must be included in S for covering the edge (r4, r5).
Advances in Greedy Algorithms by Witold Bednorz