We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems. Our techniqueproduces approximation algorithms that run in O(n² log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximationalgorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of O(n² log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n³) time for dense graphs. A similar result is obtained for the 2-matchingproblem and its variants. We also derive the first approxi...

M.I.T. We present the first polynomial-time approxima-tion algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. If k is the maximum cut requirement of the problem, our solu-tion comes within a factor of 2k of optimal. Our algo-rithm is primal-dual and shows the importance of this technique in designing approximation algorithms. 1

A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP -hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2-vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP -hard as well. We also consider the case where the graph has edge weigh...

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 k-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...

The survivable network design problem is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph

Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Edge-Connectivity Problems 3 2.1 Weighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 Vertex-Connectivity Problems 11 3.1 Weighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 Vertex-Connectivity : : : : : : : : : : : : : : : : :

We present a general approximation technique for a class of network design problems where we seek a network of minimum cost that satisfies certain communication requirements and is resilient to worst-case single-link failures. Our algorithm runs in O(n 2 log n) time on a graph with n nodes and outputs a solution of cost at most thrice the optimum. We extend our technique to obtain approximation algorithms for augmenting a given network so as to satisfy certain communication requirements and achieve resilience to single-link failures. Our technique allows one to find nearly minimum-cost two-connected networks for a variety of connectivity requirements. For example, our result generalizes earlier results on finding a minimumcost two-connected subgraph of a given edge-weighted graph in [3, 9] and an earlier result on finding a minimum-cost subgraph two-connecting a specified subset of the nodes in [14]. Using our technique, we can also approximately solve for the first time a ...

In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them.

. The MEG (minimum equivalent graph) problem is the following: "Given a directed graph, find a smallest subset of the edges that maintains all reachability relations between nodes." This problem is NP-hard; this paper gives an approximation algorithm achieving a performance guarantee of about 1:64 in polynomial time. The algorithm achieves a performance guarantee of 1:75 in the time required for transitive closure. The heart of the MEG problem is the minimum SCSS (strongly connected spanning subgraph) problem --- the MEG problem restricted to strongly connected digraphs. For the minimum SCSS problem, the paper gives a practical, nearly linear-time implementation achieving a performance guarantee of 1:75. The algorithm and its analysis are based on the simple idea of contracting long cycles. The analysis applies directly to 2-Exchange, a general "local improvement" algorithm, showing that its performance guarantee is 1:75. AMS subject classifications. 68R10, 90C27, 90C35, 05C85, 68Q20....

A Virtual Private Network (VPN) aims to emulate the services provided by a private network over the shared Internet. The endpoints of a VPN are connected using abstractions such as Virtual Channels (VCs) of ATM or Label Switching Paths (LSPs) of MPLS technologies. Reliability of an end-to-end VPN connection depends on the reliability of the links and nodes in the fixed path that it traverses in the network. In order to ensure service quality and availability in a VPN, seamless recovery from failures is essential. This work considers the problem of fast recovery in the recently proposed VPN hose model. In the hose model bandwidth is reserved for traffic aggregates instead of pairwise specifications to allow any traffic pattern among the VPN endpoints. This work assumes that the VPN endpoints are connected using a tree structure and at any time, at most one tree link can fail (i.e., single link failure model). A restoration algorithm must select asetofbackup edges and allocate necessary bandwidth on them in advance, so that the traffic disrupted by failure of a primary edge can be re-routed via backup paths. We aim at designing an optimal restoration algorithm to minimize the total bandwidth reserved on the backup edges. This problem is a variant of optimal graph augmentation problem which is NP-Complete. Thus, we present a polynomial-time approximation algorithm that guarantees a solution which is at most 16 times of the optimum. The algorithm is based on designing two reductions to convert the original problem to one of adding minimum cost edges to the VPN tree so that the resulting graph is 2-connected, which can be solved in polynomial time using known algorithms. The two reductions introduce approximation factors of 8 and 2, respectively, thus resulting in a 16-appro...