By Witold Bednorz

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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. [26] 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. [27] B. M. Waxman. “Routing of Multipoint Connections”. IEEE Journal on Selected Areas in Communications, 6(9):1617-1622, December 1988. 3 A Multilevel Greedy Algorithm for the Satisfiability Problem Noureddine Bouhmala1 and Xing Cai2 1Vestfold 2Simula University College, Research Laboratory, Norway 1.

Between every pair of nodes sj and ui for a set Qj ∈ Q and an element zi ∈ Z, the shortest path traverses through node t if zi ∈ Qj , otherwise it passes through node r. The RTs of the various nodes for of the given example above are depicted in Figure 9. Moreover, for the proof, let assume that the set of possible monitoring stations is Y = {sj│∀Qj ∈ Q} {r}. Lemma 1 Consider an instance I(Z,Q) of the SC problem and the corresponding graph (V,E) and set Y. Then there is a solution of size k to the SC problem if and only if there is a solution of size (V,E) and Y.

Thus, COST ≤ 2· COST *. Note that the time complexity of the simple probe assignment algorithm can be shown to be O(│S│·│V│2). Example 1 This example shows that the simple probe assignment algorithm has a tight approximation ratio of 2. Suppose that the cost of sending any message is 1 and consider the graph depicted in Figure 5. Let the monitoring stations be S = {s1, s2} and consider the following message assignment, A, that may be calculated by the simple algorithm. The edges (s1, s2), (s1, v1) and (vi, vi+1) of every odd i are assigned to station s1.