Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

196 S.O. Krumke et al.1. Let α =6lnn ∑ ni=1 γn i .2. From the given instance of 〈Undir, Connected, TotalP〉, construct a graphG 1(V 1,E 1) as follows.(a) For each transceiver v i (1 ≤ i ≤ n) in the problem instance, create a setg i = {u 0 i ,u 1 i ,...,u n i } of n + 1 nodes. Let the weight w(u 0 i )=α. For 1 ≤ j ≤ n,let the weight w(u j i )=γj i . The node set V1 is given by g1 ∪ g2 ∪ ...∪ gn.(b) For each i, connect the nodes in g i together as an (n + 1)-clique. (Nodes u 0 i ,1 ≤ i ≤ n, are not involved in any edges other than these clique edges.)(c) For any pair of nodes u j i and ul k, where 1 ≤ j, l ≤ n, ifγ j i ≥ p(v i,v k ) andγk l ≥ p(v k ,v i), then add the edge {u j i ,ul k} to E 1. The edge set E 1 consists ofthe clique edges added in Step 2(b) and the edges added in Step 2(c).3. Use the algorithm of [7] using a small value (say, 0.1) for ɛ to find a connecteddominating set D 1 of approximately minimal weight for G 1.4. If for some i, D 1 contains both u j i and uk i , where j