A note on the Covering Steiner Problem (original) (raw)
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Approximation algorithms for directed Steiner problems
Proceedings of the …, 1998
We give the first non-trivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ...
A greedy approximation algorithm for the group Steiner problem
Discrete Applied Mathematics, 2006
In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices {gi}i=1m. Each subset gi is called a group and the vertices in ⋃igi are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when
On subexponential running times for approximating directed Steiner tree and related problems
arXiv (Cornell University), 2018
This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1 − α) ln n, for a given parameter 0 < α < 1. What is the best possible running time for achieving such approximation ratio? This question was answered implicitly in the work of Moshkovitz [Theory of Computing, 2015]: Assuming both the Projection Games Conjecture (PGC) and the Exponential-Time Hypothesis (ETH), any ((1 − α) ln n)-approximation algorithm for Set-Cover must run in time at least 2 n c•α , for some small constant 0 < c < 1. We study the questions along this line. Our first contribution is in strengthening the above result. We show that under ETH and PGC the running time requires for any ((1 − α) ln n)-approximation algorithm for Set-Cover is essentially 2 n α. This (almost) settles the question since our lower bound matches the best known running time of 2 O(n α) for approximating Set-Cover to within a factor (1 − α) ln n given by Cygan et al. [IPL, 2009]. Our result is tight up to the constant multiplying the n α terms in the exponent. The lower bound of Set-Cover applies to all of its generalization, e.g., Group-Steiner-Tree, Directed-Steiner-Tree, Covering-Steiner-Tree and Connected-Polymatroid. We show that, surprisingly, in almost exponential running time, these problems reduce to Set-Cover. Specifically, we complement our lower bound by presenting an (1−α) ln n approximation algorithm for all aforementioned problems that runs in time 2 n α •log n • poly(m). We further study the approximation ratio in the regime of log 2−δ n for Group-Steiner-Tree and Covering-Steiner-Tree. Chekuri and Pal [FOCS, 2005] showed that Group-Steiner-Tree admits (log 2−α n)-approximation in time exp(2 log α+o(1) n), for any parameter 0 < α < 1. We show the running time lower bound of Group-Steiner-Tree: any (log 2−α n)-approximation algorithm for Group-Steiner-Tree must run in time at least exp((1 + o(1))log α−ǫ n), for any constant ǫ > 0, unless the ETH is false. Our result follows by analyzing the hardness construction of Group-Steiner-Tree due to the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for Covering-Steiner-Tree.
An algorithm f o r solving the Steiner problem on a f i n i t e undirected graph i s presented. s e t of graph arcs of minimum t o t a l length needed t o connect a specified s e t of k graph nodes. I f the entire graph contains n nodes, the algorithm requires time proportional t o ThiB algorithm computes the 3 The t h e requirement above includes the term n / 2 , which can be eliminated i f the s e t of shortest paths connecting each pair of nodes in the graph i s available. Also, the RAND Corporation, which through the U.S. A i r Force under t h e Project RAND, supported the authors during t h e i r i n i t i a l research on t h i s problem. i n p a r t i s permitted for any purposes of the United S t a t e s Government.
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.
On-line generalized Steiner problem
Theoretical Computer Science, 2004
The generalized Steiner problem (GSP) is deÿned as follows. We are given a graph with non-negative edge weights and a set of pairs of vertices. The algorithm has to construct minimum weight subgraph such that the two nodes of each pair are connected by a path.
The Steiner problem with edge lengths 1 and 2
Information Processing Letters, 1989
The Steiner problem on networks asks for a shortest subgraph spanning a given subset of distinguished vertices. We give a !-approximation algorithm for the special case in which the underlying network is complete and all edge lengths are either 1 or 2. We also relate the Steiner problem to a complexity class recently defined by Papadimitriou and Yannakakis by showing that this special case is MAX SNP-hard, which may be evidence that the Steiner problem on networks has no polynomial-time approximation scheme.
Approximating Rooted Steiner Networks
ACM Transactions on Algorithms, 2014
The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques (due to others), we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω(k ) hardness bound for the rooted k-connectivity problem in undirected graphs; this addresses a recent open question of Khanna. As a consequence, we also obtain the Ω(k ) hardness of the undirected subset k-connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted kconnectivity problem.
Approximating Fault-Tolerant Group-Steiner problems
2009
In this paper, we initiate the study of designing approximation algorithms for Fault-Tolerant Group-Steiner (FTGS) problems. The motivation is to protect the well-studied group-Steiner networks from edge or vertex failures. In Fault-Tolerant Group-Steiner problems, we are given a graph with edge-(or vertex-) costs, a root vertex, and a collection of subsets of vertices called groups. The objective is to find a minimum-cost subgraph that has two edge-(or vertex-) disjoint paths from each group to the root. We present approximation algorithms and hardness results for several variants of this basic problem, e.g., edge-costs vs. vertex-costs, edge-connectivity vs. vertex-connectivity, and 2-connecting a single vertex vs. two distinct vertices from each group. Main contributions of our paper include the introduction of general structural lemmas on connectivity and a charging scheme that may find more applications in the future. Our algorithmic results are supplemented by inapproximability results, which are tight in some cases.