Power assignment problems take as input a directed simple graph G = (V;E) and a cost function c : E ! R+. A solution to this problem assigns... Show morePower assignment problems take as input a directed simple graph G = (V;E) and a cost function c : E ! R+. A solution to this problem assigns every vertex a nonnegative power, p(v). We use H = (V;B(p)) to denote the spanning subgraph of G created by this power assignment. Let B(p) denote the set of all the links established between pairs of nodes in V under the power assignment p. The minimization problem then is to find the minimum power assignment, Pp(v), subject to H satisfying a specific property. 4 variants of this problem are discussed in this paper (a) Min-Power Strong Connectivity: H = (V;B(p)) is strongly connected. (b) Min-Power Broadcast: H = (V;B(p)) has a path from the fixed source z to every other vertex. (c) Min-Power Connectivity with 2-level power (Symmetric): c : E ! f0; 1g and H = (V;B(p)) is connected. (d) Min- Power Strong Connectivity with 2-level power (Asymmetric): c : E ! f0; 1g and H = (V;B(p)) is strongly connected. We give the exact solution using an improved integer linear program for problem (a) and (b) (We do not have a section for the integer linear program of Min-Power Broadcast problem since it is very similar to Min-Power Strong connectivity). Then we try to speedup current best approximation algorithms while preserving their approximation ratio. For problem (a), we give a fast variant of 1:85-approximation algorithm with running time O(n2 log2 n). For problem (b), we give a fast variant of 2(1 + ln n)-approximation algorithm for the most general cost model with running time O(n3) and a fast variant of 4:2- approximation algorithm for 2-dimensional cost model with running time O(nm), where n = jV j and m = jEj. For both problem (c) and (d), We give 5 3-approximation algorithms that run in O(m (n)), where (n) is the inverse Ackermann function. Ph.D. in Computer Science, May 2015 Show less
