Approximating Minimum Power Covers of Intersecting Families and Directed Connectivity Problems

Abstract

Given a (directed) 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 fundamental directed connectivity network design problems under the power minimization criteria. Let G = (V, E) be a graph with edge-costs {c(e) : e ∈ E} and let k be an integer. We consider finding a minimum power subgraph G of G that satisfies some prescribed property. The Min-Power k-Outconnected Subgraph (MPk-OS) problem requires that G contains k pairwise internally-disjoint rv-paths for all v ∈ V − r, for a given "root" r ∈ V . The Min-Power k-Connected Subgraph (MPk-CS) problem requires that G contains k pairwise internally-disjoint paths between every pair of its nodes. In the edge connectivity variant the paths are required to be only edge disjoint. For k = 1 all these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln |V |) approximation threshold. For k = Θ(n) the edge connectivity variants are unlikely to admit a polylogarithmic approximation algorithm [23]. We give an O(k ln |V |)-approximation algorithms for these four problems. Our algorithms are based on a much more general O(ln |V |)-approximation algorithm for the problem of finding a min-power directed (edge-)cover of an intersecting set-family; a set-family F is intersecting if X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F, and an edge set F covers F if for every X ∈ F there is an edge in F entering X.

Approximation Algorithms for Disjoint st-Paths with Minimum Activation Cost

“…Theorem 2 thus implies that there is an exact polynomial-time algorithm for the directed MA2EDP problem for power optimization, as was already shown in [12]. Nutov [8] shows that for arbitrary k there exists a k-approximation algorithm for the directed case of the MAkEDP problem.…”

“…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 .…”

Approximating Minimum-Power k-Connectivity

“…For further results on other min-power connectivity problems, among them problems on directed graphs see [9,21,17]. For results on min-cost k-connectivity problems see [2,6,14,5,15,7,20,18]; see also a recent survey in [16] on various min-cost connectivity problems.…”

“…Except directed MPkDP and MPkIS that are in P, there is still a large gap between upper and lower bounds of approximation for many other min-power node connectivity problems, for both directed and undirected graphs, see [21,17,12].…”

“…Another example is finding an arborescence rooted at s, that is, a subgraph that contains an sv-path for every node v. The minimum-cost case is in P (c.f., [22]), while the minimum-power variant is at least as hard as the Set-Cover problem. For more examples see [1,21].…”