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We next show that if there exists a feasible probe assignment A containing at most 5·│ │+│ │+k probes, then there exists a vertex cover of size at most k for . Let V’ consist of all nodes such that A contains the probe (ui3,wi). Since each edge (wi,wj) is in the RT of only ui3 or uj3, A must contain one of (ui3,wi) or (uj3,wj ), and thus V’ must be a vertex cover of . Further, we can show that V’ contains at most k nodes. Suppose that this is not the case and V’ contains more than k nodes. Then, A must contain greater than k probes (ui3,wi) for ∈ .

In Proceedings of the IEEE INFOCOM’2003, San Francisco, CA, USA, April 2003. [11] V. Chavatel, “A Greedy Heuristic for the Set-Covering Problem”, Math. of Operation Research, Vol. 4, No. 3, pp 233-235, 1979. [12] K. Moore, “SONAR - A Network Proximity Service, Version 1”. org/utk/projects/sonar/ August 1998. [13] P. Francis, S. Jamin, V. Paxson, L. Zhang, D. F. Gryniewicz, and Y. Jin, “An Architecture for a Global Internet Host Distance Estimation Service”, In Proceedings of IEEE INFOCOM'99, New York City, New York, March 1999.

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). Suppose without lose of generality that the selected nodes are r2 and r4. Let us turn to describe the different SPTs of the nodes in G ( V , E ). The SPTs and are very similar. The SPT contains the edge (r2, r3) and all the incident edges of node r1 except edge (r1, r3). The SPT contains the edge (r4, r5) and all the incident edges of node r1 except edge (r1, r5).

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