On minimum power connectivity problems

Abstract

Given a (directed or undirected) graph with edge costs, 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 present polynomial and improved approximation algorithms, as well as inapproximability results, for some classic network design problems under the power minimization criteria. Our main result is for the problem of finding a min-power subgraph that contains k internally-disjoint vs-paths from every node v to a given node s: we give a polynomial algorithm for directed graphs and a logarithmic approximation algorithm for undirected graphs. In contrast, we will show that the corresponding edge-connectivity problems are unlikely to admit even a polylogarithmic approximation.

Approximating Minimum Power Edge-Multi-Covers

“…The previous best ratio for MPkOS was O(log n) + 4 = O(log n) [5] for large values of k = Ω(log n), and k + 1 for small values of k [9]. From Theorem 1 we obtain the following.…”

“…In [5] it is proved that an α-approximation for MPEMC implies an (α + 4)-approximation for MPkOS. The previous best ratio for MPkOS was O(log n) + 4 = O(log n) [5] for large values of k = Ω(log n), and k + 1 for small values of k [9].…”

Minimum Activation Cost Node-Disjoint Paths in Graphs with Bounded Treewidth

“…In the undirected case we say the edge (u, v) is activated for chosen values x u and x v if each of these values is at least θ uv , and in the directed case it is activated if x u ≥ θ uv . [7] shows that a simple reduction to the shortest st-path problem can solve the minimum power st-path (MPP) problem in polynomial time for directed/undirected networks. The problem of finding minimum power k node-disjoint st-paths (st-MPkNDP) in directed graphs can be solved in polynomial time [6,13].…”

Approximation Algorithms for Disjoint st-Paths with Minimum Activation Cost

“…Other relevant work has addressed power optimization [2,7,8]. In power optimization problems, each edge (u, v) ∈ E has a threshold power requirement θ uv .…”

“…As mentioned in [11], in the power optimization setting the MEAN and MNAN problems have 4-approximation and 11/3-approximation algorithms, respectively, and it is known that the MSpAT problem is APX-hard. By a simple reduction to the shortest st-path problem, the Minimum Power st-Path problem is solvable in polynomial time for both directed and undirected networks [7]. Another problem that has been studied in the literature is finding the Minimum Power k Edge-Disjoint st-Paths (MPkEDP).…”