Minimum Power Connectivity Problems

Given a (directed or undirected) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications for wireless networks, we consider some fundamental network design problems under the power minimization criteria. Let G = (V, E) be a graph with edge-costs {ce : e ∈ E} and let k be an integer. We consider finding a min-power subgraph G of G that satisfies some prescribed connectivity requirements. The Min-Power k Edge-Disjoint Paths (MPk-EDP) problem requires that G contains k pairwise edge-disjoint st-paths for given s, t ∈ V ; the Min-Power k-Edge-Outconnected Subgraph (MPk-EOS) problem requires that G contains k pairwise edge-disjoint sv-paths for all v ∈ V − s, for given s ∈ V ; and the Min-Power k-Edge-Connected Subgraph (MPk-ECS) problem requires that G is spanning and k-connected. When the paths are required to be internally disjoint, we get the problems Min-Power k Disjoint Paths (MPk-DP), Min-Power k-Outconnected Subgraph (MPk-OS), and Min-Power k-Connected Subgraph (MPk-CS), respectively. We survey the currently best known approximation algorithms for these problems, mainly for directed graphs. We then present our original results as follows. We give an evidence that the undirected MPk-EDP and MPk-ECS and directed MPk-EOS and MPk-ECS are unlikely to admit a polylogarithmic approximation ratio even for unit costs. On the other hand, for both directed and undirected graphs we give a polynomial time algorithm for finding a min-power augmenting edge set that increases the st-edge-connectivity by 1; this implies a k-approximation algorithm for undirected MPk-EDP. We also give a min{k + 4, O(log n)}-approximation algorithm for node-connectivity version of undirected MPk-EOS; this improves the previously best known ratio of 2k − 1/3.