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...

We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair fi; jg of nodes, an edge-connectivity requirement r ij . The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

The Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-time heuristic with an approximation ratio approaching 1 + 2 1:55, which improves upon the previously best-known approximation algorithm of [10] with performance ratio 1:59.

... problem is very important for such aspects of physical layout as global routing and wiring estimation. Virtually all previous heuristics for computing rectilinear Steiner trees begin with a minimum spanning tree (MST) topology and rearrange edges to induce Steiner points. This paper gives a more direct approach which makes a significant departure from such spanning treebased strategies: we iteratively find optimal Steiner points to be added to the layout. Our method not only gives improved average-case performance, but also escapes the worst-case examples of existing approaches. We show that the performance ratio of our method can never be as bad as 3/2, and is in fact bounded by 4/3 on the entire class of instances where the c(MST)/c(MRST) cost ratio is exactly 3/2. Sophisticated computational geometry techniques allow efficient and practical implementation, and the method is naturally suited to technological regimes where, e.g., via costs can be high and the number of Steiner points should be limited. Extensive performance results show a 2 % to 3 % wire length reduction over the best previous heuristics. We describe a number of variants and extensions, and suggest directions for further research.

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

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

Abstract. The classical Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present a new polynomial-ln 3 time heuristic that achieves a best-known approximation ratio of 1 + ≈ 1.55 for general graphs 2 and best-known approximation ratios of ≈ 1.28 for both quasi-bipartite graphs (i.e., where no two nonterminals are adjacent) and complete graphs with edge weights 1 and 2. Our method is considerably simpler and easier to implement than previous approaches. We also prove the first known nontrivial performance bound (1.5 · OPT) for the iterated 1-Steiner heuristic of Kahng and Robins in quasi-bipartite graphs.

We study network-design problems with multiple design objectives. In particular, we look at two cost measures to be minimized simultaneously: the total cost of the network and the maximum degree of any node in the network. Our main result can be roughly stated as follows: given an integer b, we present approximation algorithms for a variety of network-design problems on an n- node graph in which the degree of the output network is O(b log( n b )) and the cost of this network is O(log n) times that of the minimum-cost degree-b-bounded network. These algorithms can handle costs on nodes as well as edges. Moreover, we can construct such networks so as to satisfy a variety of connectivity specifications including spanning trees, Steiner trees and generalized Steiner forests. The performance guarantee on the cost of the output network is nearly best-possible unless NP = ~ P . We also address the special case in which the costs obey the triangle inequality. In this case, we obtai...

The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G = (V; E; d) and Euclidean plane, respectively. For these problems, we present a series of heuristics finding approximate Steiner tree with performance guarantee coming arbitrary close to 1+ln2 1:693 and 1+ln 2 p 3 1:1438, respectively. The best previously known corresponding values are close to 1.746 and 1.1546. Keywords: Combinatorial problems, approximation algorithms, Steiner trees. 1 Introduction Let G = (V; E;d) be a graph with a vertex set V , an edge set E and distance function d : E ! R + . A tree T is called a Steiner tree of S, S ae V , if S is contained in the vertex set of T . Network Steiner Problem (NSP). Given G and S, find the shortest Steiner tree (also called the Steiner minimal tree) of S. This problem is NP-complete [9], so many approximation algorithms for Steiner minimal trees appeared in the last two decades. The quality of an appr...

The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NP-hard, and the best performing MRST heuristic to date is the Iterated 1-Steiner (I1S) method recently proposed by Kahng and Robins. In this paper we develop a straightforward, efficient implementation of I1S, achieving a speedup factor of three orders of magnitude over previous implementations. We also give a parallel implementation that achieves near-linear speedup on multiple processors. Several performance-improving enhancements enable us to obtain Steiner trees with average cost within 0.25% of optimal, and our methods produce optimal solutions in up to 90% of the cases for typical nets. We generalize I1S and its variants to three dimensions, as well as to the case where all the pins lie on k parallel planes, which arises in, e.g., multi-layer routing. Motivated by the goal of reducing the running times of our algorith...