A capacitated covering IP is an integer program of the form min{cx|Ux ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as ||d||∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of non-zero coefficients in a row of U, and provide a polynomial-time approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first non-trivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomial-time approximation algorithms to match this bound. This improves on the best previous m-approximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentially-many) inequalities added. Our contribution here is to describe an appropriate

Abstract: An efficient heuristic is presented for the problem of finding a minimum-size k- connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimum-size k-node connected spanning subgraph of an undirected graph 1+[1=k], minimum-size k-node connected spanning subgraph of a directed graph 1+[1=k], minimum-size k-edge connected spanning subgraph of an undirected graph 1+[2=(k + 1)], and minimum-size k-edge connected spanning subgraph of a directed graph 1+[4= p k].

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 problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G = (V; E) is considered. For k 2, this problem is known to be NP-hard. Combining properties of inclusionminimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a (d k 2 e+1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(jV j 3 jEj) = O(jV j 5 ). Up to 1990, E. A. Dinic, Moscow. y Dept. of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. E-mail: dinitz@cs.bgu.ac.il. z This work was done as a part of the author's D.Sc. thesis at the ...

We present the first truly polynomial-time approximation scheme (PTAS) for the minimum-cost k-vertex- (or, k- edge-) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " how to construct in a polynomial time a graph on a superset of the input points which is k-vertex connected with respect to the input points, and whose cost is within (1+ ") of the minimum-cost of a k-vertex connected graph spanning the input points. We subsume that result by showing for every positive constant " how to construct in a polynomial-time a k-connected subgraph spanning the input points without any Steiner points and having the cost within (1 + ") of the minimum. We also study hardness of approximations for the minimum-cost k-vertex- and k-edge-connected spanning subgraph problems. The only inapproximability result known so far for the minimum-cost k-vertex- and k-edge- connected spanning subgraph problems states that the k- edge...

We introduce a new algorithmic technique that applies to several graph connectivity problems. Its power is demonstrated by experimental studies of the minimum-weight strongly-connected spanning subgraph problem and the minimum-weight augmentation problem. Even though we are unable to improve the approximation ratios for these problems, our studies indicate that the new method generates significantly better solutions than the current known approximation algorithms, and yields solutions very close to optimal. We believe that our technique will eventually lead to algorithms that improve the performance ratios as well. 1 Introduction Let a weighted graph G = (V; E) represent all the feasible links of a potential communications network. A minimum spanning tree in G is the cheapest connected subgraph, i.e., the cheapest network that will allow the sites to communicate. Such a network is highly susceptible to failures, since it cannot even survive a single link or site failure. For more rel...

. Let H be a laminar family of subsets of a groundset V . A k-cover of H is a multiset C of edges on V such that for every subset S in H, C has at least k edges that have exactly one end in S. A k-packing of H is a multiset P of edges on V such that for every subset S in H, P has at most k \Delta u(S) edges that have exactly one end in S. Here, u assigns an integer capacity to each subset in H. Our main results are: (a) Given a k-cover C of H, there is an efficient algorithm to find a 1-cover contained in C of size kjCj=(2k \Gamma 1). For 2-covers, the factor of 2=3 is best possible. (b) Given a 2-packing P of H, there is an efficient algorithm to find a 1-packing contained in P of size jP j=3. The factor of 1=3 for 2-packings is best possible. These results are based on efficient algorithms for finding appropriate colorings of the edges in a k-cover or a 2-packing, respectively, and they extend to the case where the edges have nonnegative weights. Our results imply app...

We present new polynomial-time approximation schemes (PTAS) for several basic minimum-cost multi-connectivity problems in geometrical graphs. We focus on low connectivity requirements. Each of our schemes either significantly improves the previously known upper time-bound or is the first PTAS for the considered problem. We provide a randomized approximation scheme for finding a biconnected graph spanning a set of points in a multi-dimensional Euclidean space and having the expected total cost within (1+ε) of the optimum. For any constant dimension and ε, our scheme runs in time O(n log n). It can be turned into Las Vegas one without affecting its asymptotic time complexity, and also efficiently derandomized. The only previously known truly polynomial-time approximation (randomized) scheme for this problem runs in expected time n · (log n) O((log log n)9) in the simplest planar case. The efficiency of our scheme relies on transformations of nearly optimal low cost special spanners into sub-multigraphs having good decomposition

This paper considers the problem of augmenting a given graph by a cheapest possible set of additional edges in order to make the graph edge-biconnected. An application is the extension of an existing communication network to become robust against single link-failures. An evolutionary algorithm (EA) is presented which includes an e#ective preprocessing of the problem data and a local improvement procedure that is applied during initialization, recombination, and mutation. In this way, the EA searches the space of feasible, locally optimal solutions only. The variation operators were designed with particular emphasis on low computational e#ort and strong locality. Empirical results indicate the superiority of the new approach over two previous heuristic methods

The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uv-edge-connectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jain [19], and nume-rous papers preceding it, considered the Edge-Weighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum weight edge-covers of several types of set functions and families. However, for the Node-Weighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation, by giving the first non-trivial approxi-mation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(ln |U|), where rmax = maxu,v∈U r(u, v). This generalizes the result of Klein and Ravi [21] for the case rmax = 1. We also give an O(ln |U|)-approximation algorithm for the node-connectivity variant of NWSN (when the paths are required to be internally-disjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum node-weighted edge-cover of an uncrossable set-family. We also give the first evidence that a polylogarithmic approximation ratio for NWSN might not exist even for |U | = 2 and unit weights. 1 1