We give approximation algorithms for the Survivable Network problem. The input consists of a graph G = ( V,E ) with edge/node-costs, a node subset S ⊆ V , and connectivity requirements { r ( s,t ): s,t ∈ T ⊆ V }. The goal is to find a minimum cost subgraph H of G that for all s,t ∈ T contains r ( s,t ) pairwise edge-disjoint st -paths such that no two of them have a node in S ∖ { s,t } in common. Three extensively studied particular cases are: Edge-Connectivity Survivable Network ( S = ∅), Node-Connectivity Survivable Network ( S = V ), and Element-Connectivity Survivable Network ( r ( s,t ) = 0 whenever s ∈ S or t ∈ S ). Let k = max s,t ∈ T r ( s,t ). In Rooted Survivable Network, there is s ∈ T such that r ( u,t ) = 0 for all u ≠ s , and in the Subset k -Connected Subgraph problem r ( s,t ) = k for all s,t ∈ T . For edge-costs, our ratios are O ( k log k ) for Rooted Survivable Network and O ( k 2 log k ) for Subset k -Connected Subgraph. This improves the previous ratio O ( k 2 log n ), and for constant values of k settles the approximability of these problems to a constant. For node-costs, our ratios are as follows. — O ( k log | T |) for Element-Connectivity Survivable Network, matching the best known ratio for Edge-Connectivity Survivable Network. — O ( k 2 log | T |) for Rooted Survivable Network and O ( k 3 log | T |) for Subset k -Connected Subgraph, improving the ratio O ( k 8 log 2 | T |). — O ( k 4 log 2 | T |) for Survivable Network; this is the first nontrivial approximation algorithm for the node-costs version of the problem.
We present a 1.5-approximation algorithm for the following NP-hard problem: given a connected graph G = (V, E) and an edge set E on V disjoint to E, find a minimum size subset of edges F ⊆ E such that (V, E ∪ F ) is 2-edge-connected. Our result improves and significantly simplifies the approximation algorithm with ratio 1.875 + ε of Nagamochi.
Abstract. Given a 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 the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-ECSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log 4 n)-approximation algorithm for MPb-EC. This gives an O(log 4 n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an O( √ n) approximation algorithm for MPk-ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2 log 1−ε n ) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time. IntroductionWireless multihop networks are an important subject of study due to their extensive applications (see e.g., [8,24] performing network tasks while minimizing the power consumption of the radio transmitters in the network. In ad-hoc networks, a range assignment to radio transmitters means to assign a set of powers to mobile devices. We consider finding a range assignment for the nodes of a network such that the resulting communication network satisfies some prescribed properties, and such that the total power is minimized. Specifically, we consider "min-power" variants of three extensively studied "min-cost" problems: the b-Edge Cover problem and the k-Connected Spanning Subgraph Problem in undirected networks, and the k-Edge-Disjoint Paths problem in directed networks.In wired networks, generally we want to find a subgraph with the minimum cost instead of the minimum power. This is the main difference between the optimization problems for wired versus wireless networks. The power model for undirected graphs corresponds to static symmetric multi-hop ad-hoc wireless networks with omnidirectional transmitters. This model is justified and used in several other papers [3,4,14].An important network task is assuring high fault-tolerance ( [1-4, 11, 18]).The simplest version is when we require the network to be connected. In this case, the min-cost variant is just the min-cost spanning tree problem, while the min-power variant is NP-hard even in the Euclidean plane [9]. There are several localized and distributed heuristics to find the range assignment to keep the network connected [18,24,25]. Constant approximation guarantees for the min-power spanning tree problem are given in [4,14]. For general k, the bestPower Optimization for Connectivity Problems 3 previously known approximation ratio fo...
Given a graph (directed or undirected) with costs on the edges, and an integer k, we consider the problem of finding a k-node connected spanning subgraph of minimum cost. For the general instance of the problem (directed or undirected), there is a simple 2k-approximation algorithm. Better algorithms are known for various ranges of n, k. For undirected graphs with metric costs Khuller and Raghavachari gave a 2 + 2(k−1) n -approximation algorithm. We obtain the following results.(i) For arbitrary costs, a k-approximation algorithm for undirected graphs and a (k + 1)-approximation algorithm for directed graphs.(ii) For metric costs, a (2 + k−1 n )-approximation algorithm for undirected graphs and a (2 + k n )-approximation algorithm for directed graphs. For undirected graphs and k = 6, 7, we further improve the approximation ratio from k to (k + 1)/2 = 4; previously, (k + 1)/2 -approximation algorithms were known only for k ≤ 5. We also give a fast 3-approximation algorithm for k = 4.The multiroot problem generalizes the min-cost k-connected subgraph problem. In the multiroot problem, requirements k u for every node u are given, and the aim is to find a minimum-cost subgraph that contains max{k u , k v } internally disjoint paths between every pair of nodes u, v. For the general instance of the problem, the best known algorithm has approximation ratio 2k, where k = max k u . For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve for k ≤ 7 the approximation guarantee from 3 to 2+ (k−1)/2 k < 2.5.
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