Approximating minimum cost connectivity problems (original) (raw)
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Hardness of Approximation for Vertex-Connectivity Network Design Problems
SIAM Journal on Computing, 2004
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.
Approximating Node Connectivity Problems via Set Covers
Algorithmica, 2003
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:
Set connectivity problems in undirected graphs and the directed Steiner network problem
ACM Transactions on Algorithms, 2011
In the generalized connectivity problem, we are given an edge-weighted graph G = (V, E) and a collection D = {(S 1 , T 1 ), . . . , (S k , T k )} of distinct demands; each demand (S i , T i ) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand (S i , T i ) when it contains a path with one endpoint in S i and the other in T i . The goal is to identify a minimum weight subgraph that connects all demands in D. Alon et al. (SODA '04) introduced this problem to study online network formation settings and showed that it captures some wellstudied problems such as Steiner forest, facility location with non-metric costs, tree multicast, and group Steiner tree. Finding a non-trivial approximation ratio for generalized connectivity was left as an open problem. We describe the first poly-logarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O(log 2 n log 2 k). Here, n is the number of vertices in G and k is the number of demands. We also prove that the cut-covering relaxation of this problem has an O(log 3 n log 2 k) integrality gap.
An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem
Algorithmica, 2014
The minimum cost subset k-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph G = (V, E) with costs on edges and a set of terminals T. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by k openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset k-connected subgraph problem which improves on the previous best approximation guarantee of O(k 2 log k) by Nutov (FOCS 2009). Our approximation guarantee, α(|T |), depends upon the number of terminals: α(|T |) = O(k log 2 k) if 2k ≤ |T | < k 2 O(k log k) if |T | ≥ k 2 So, when the number of terminals is large enough, the approximation guarantee improves significantly. Moreover, we show that, given an approximation algorithm for |T | = k, we can obtain almost the same approximation guarantee for any instances with |T | > k. This suggests that the hardest instances of the problem are when |T | ≈ k.
We design combinatorial approximation algorithms for the Capacitated Steiner Network (Cap-SN) problem and the Capacitated Multicommodity Flow (Cap-MCF) problem. These two problems entail satisfying connectivity requirements when edges have costs and hard capacities. In Cap-SN, the flow has to be supported separately for each commodity while in Cap-MCF, the flow has to be sent simultaneously for all commodities. We show that the Group Steiner problem on trees ([12]) is a special case of both problems. This implies the first polylogarithmic lower bound for these problems by [17]. We then give various approximations to special cases of the problems. We generalize the well known Source location problem (see for example [19]), to a natural problem called the Connected Rent or Buy Source Location problem. We show that this problem is a a simplification of Cap-SN and Cap-MCF and a generalization of Group Steiner on general graphs. We use Group Steiner Tree techniques, and more sophisticated techniques, to derive log 3+ n approximation for the Connected Rent or Buy Source Location problem which is close to the best approximation known for Group Steiner on general graphs. Another special case we study is as follows. Given a bipartite graph G = (A ∪ B, E) and an integer k > 0, find A ⊆ A and B ⊆ B of minimum total size |A | + |B | such that there exist k edge-disjoint paths in G from vertices in A to vertices in B. This problem is a special case of the Steiner Network problem with vertex costs [20]. In [20] Nutov asked the open question if the Steiner network problem with vertex costs admits an o(k) ratio. We give an o(k) approximation for this special case, which could be a step toward resolving the open question of Nutov. We provide an O(√ k log k) approximation ratio for the problem. We also show that we can compute a solution of optimum value, while being able to route O(k/polylog n) flow, where n is Part of this work was done at DIMACS. We thank DIMACS for their hospitality.
Approximating some network design problems with node costs
Theoretical Computer Science, 2011
We study several multi-criteria undirected network design problems with node costs and lengths. All these problems are related to the Multicommodity Buy at Bulk (MBB) problem in which we are given a graph G = (V , E), demands {d st : s, t ∈ V }, and a family {c v : v ∈ V } of subadditive cost functions. For every s, t ∈ V we seek to send d st flow units from s to t, so
Better algorithms for minimum weight vertex-connectivity problems
Lecture Notes in Computer Science, 1997
Given a k vertex connected graph with weighted edges, we study the problem of nding a minimum weight spanning subgraph which is k vertex-connected, for k = 2; 3; 4. The problem is known to be NP-hard for any k 2, even when edges have no weight.
A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph
Journal of Algorithms, 1999
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 ( k 2 + 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(|V | 3 |E|) = O(|V | 5 ). * Up to 1990,