Approximation Algorithms for (S,T)-Connectivity Problems

We study a directed network design problem called the $k$-$(S,T)$-connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer $k$, the minimum cost $k$-vertex connected spanning subgraph problem is a special case of the $k$-$(S,T)$-connectivity problem. We defer precise statements of the problem and of our results to the introduction. For $k=1$, we call the problem the $(S,T)$-connectivity problem. We study three variants of the problem: the standard $(S,T)$-connectivity problem, the relaxed $(S,T)$-connectivity problem, and the unrestricted $(S,T)$-connectivity problem. We give hardness results for these three variants. We design a $2$-approximation algorithm for the standard $(S,T)$-connectivity problem. We design tight approximation algorithms for the relaxed $(S,T)$-connectivity problem and one of its special cases. For any $k$, we give an $O(\log k\log n)$-approximation algorithm, where $n$ denotes the number of vertices. The approximation guarantee almost matches the best approximation guarantee known for the minimum cost $k$-vertex connected spanning subgraph problem which is $O(\log k\log\frac{n}{n-k})$ due to Nutov in 2009

Improved Approximation Algorthmsor Uniform Connectivity Problems

The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NP-hard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. The following results are presented: 1. For the unweighted k-edge-connectivity problem an approximation algorithm that achieves a performance ratio of 1.85 is described. This is the first polynomial-time algorithm that achieves a constant less than 2, for all k. 2. For the weighted vertex-connectivity problem, a constant factor approximation algorithm is given assuming that the edge-weights satisfy the triangle inequality. This is the first constant factor approximation algorithm for this problem. 3. For the case of biconnectivity, with no assumptions about the weights of the edges, an algorithm that achieves a factor asymptotically approaching 2 is described. This matches the previous best bound for the corresponding edge connectivity problem. (Also cross-referenced as UMIACS-TR-95-21

A fault-tolerant relay placement algorithm for ensuring k vertex-disjoint shortest paths in wireless sensor networks

Wireless sensor networks (WSNs) are prone to failures. To be robust to failures, the network topology should provide alternative routes to the sinks so when failures occur the routing protocol can still offer reliable delivery. Our contribution is a solution that enables fault-tolerant WSN deployment planning by judicious use of a minimum number of additional relays. A WSN is robust if at least one route with an acceptable length to a sink is available for each sensor node after the failure of any nodes. In this paper, we define the problem for increasing WSN reliability by deploying a number of additional relays to ensure that each sensor node in the initial design has k length-bounded vertex-disjoint shortest paths to the sinks. To identify the maximum k such that each node has k vertex-disjoint shortest paths, we propose Counting-Paths and its dynamic programming variant. Then, we introduce GRASP-ARP, a centralised offline algorithm that uses Counting-Paths to minimise the number of deployed relays. Empirically, it deploys 35% fewer relays with reasonable runtime compared to the closest approach. Using network simulation, we show that GRASP-ARP’s designs offer a substantial improvement over the original topologies, maintaining connectivity for twice as many surviving nodes after 10% of the original nodes have failed

Approximation Algorithms for Finding Highly Connected Subgraphs

(Also cross-referenced as UMIACS-TR-95-4

Augmenting Trees to Achieve 2-Node-Connectivity

This thesis focuses on the Node-Connectivity Tree Augmentation Problem (NC-TAP), formally defined as follows. The first input of the problem is a graph G which has vertex set V and edge set E. We require |V| >= 3 to avoid degenerate cases. The edge set E is a disjoint union of two sets T and L where the subgraph (V,T) is connected and acyclic. We call the edges in T the tree edges and the edges in L are called links. The second input is a vector c in R^L, c >= 0 (a vector of nonnegative real numbers indexed by the links), which is called the cost of the links. We often refer to this graph G and cost vector c as an instance of NC-TAP. Given an instance G = (V, T U L) and c to NC-TAP, a feasible solution to that instance is a set of links F such that the graph (V, T U F) is 2-connected. The cost of a set of links. The goal of NC-TAP is to find a feasible solution F^* to the given instance such that the the cost of F^* is minimum among all feasible solutions to the instance. This thesis is mainly expository and it has two goals. First, we present the current best-known algorithms for NC-TAP. The second goal of this thesis is to explore new directions in the study of NC-TAP in the last chapter. This is an exploratory chapter where the goal is to use the state of the art techniques for TAP to develop an algorithm for NC-TAP which has an approximation guarantee better than factor 2

Assessing the Connectivity Reliability of a Maritime Transport Network: A Case of Imported Crude Oil in China

Crude oil transportation is a vital component of the global energy supply, and the global Crude Oil Maritime Transportation Network (COMTN) plays a crucial role as a carrier for crude oil transportation. Once the network faces attacks that result in the failure of certain routes, a severe threat is posed to the crude oil supply security of importing countries. Therefore, it is crucial to evaluate the reliability of the COMTN. This study proposes a model for evaluating the reliability of the imported COMTN by analyzing the impact of node failures. Firstly, the network is constructed using complex networks (CNs) theory, with ports, canals, and straits as nodes, and shipping routes as directed edges. Secondly, based on the Weighted Leader Rank algorithm, a comprehensive evaluation metric for CNs is established, and a node importance assessment model is developed to rank the nodes accordingly. Thirdly, a case study is conducted using China’s imported COMTN as an example, evaluating the connectivity reliability (CR) under random and deliberate attack scenarios. Finally, measures and recommendations are provided to enhance the CR of China’s imported COMTN. The findings indicate that deliberate attacks pose a greater threat, and reliability varies across maritime routes, with the Americas route exhibiting higher reliability compared to the Middle East and Southeast Asia routes. The results of this study can provide relevant recommendations for policy makers. The model proposed in this study can also be applied to other countries and regions to assess the connectivity reliability of their local COMTNs and develop appropriate measures for the results