Approximation algorithms for directed Steiner problems (original) (raw)
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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.
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.
Improved Approximation Algorithms for the Quality of Service Steiner Tree Problem
2003
The Quality of Service Multicast Tree Problem is a generalization of the Steiner tree problem which appears in the context of multimedia multicast and network design. In this generalization, each node possesses a rate and the cost of an edge with length l in a Steiner tree T connecting the source to non-zero rate nodes is l · re, where re is the maximum node rate in the component of T − {e} that does not contain the source. The best previously known approximation ratios for this problem (based on the best known approximation factor of 1.549 for the Steiner tree problem in networks) are 2.066 for the case of two non-zero rates and 4.212 for the case of unbounded number of rates. In this paper we give improved approximation algorithms with ratios of 1.960 and 3.802, respectively. When the minimum spanning tree heuristic is used for finding approximate Steiner trees, then the previously best known approximation ratios of 2.667 for two non-zero rates and 5.542 for unbounded number of rates are reduced to 2.414 and 4.311, respectively.
2-Approximation Algorithm for the Minimum Weighted Steiner Tree Problem
The Minimum Spanning Tree problem is well-known and has been studied extensively. The solution to this problem spans all vertices of a graph. Nonetheless, a more generalized problem-the Steiner Minimal Tree problem-is yet to be delved into thoroughly. The solution to this problem spans only a required subset of vertices of a graph. In this paper, we describe several algorithms to solve the Steiner Minimal Tree problem, and investigate specifically how the the Steiner Minimal Tree problem can be solved using a 2-approximation algorithm, with an application in essentially large instances. Due to the NP-hardness of the problem, approximation algorithms prove to be the sole feasible solution, unless the given instances are exceptionally small. We also discuss data structures that optimize the approximation algorithm for large graphs, and evaluate the time complexity of our implementation of the algorithm.
Differential approximation results for the Steiner tree problem
we study the approximability of three versions of the Steiner tree problem. For the first one where the input graph is only supposed connected, we show that it is not approximable within better than IV \ Nj-' for any E E (0, l), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where the input graph is supposed complete and the edge distances are arbitrary, we prove that it can be differentially approximated within l/2. For the third one defined on complete graphs with edge distances 1 or 2, we show that it is differentially approximable within 0.82. Also, extending the result of Bern and Plassmann [l], we show that the Steiner tree problem with edge lengths 1 and 2 is MaxSNP-complete even in the case where IV1 < TINI, for any T > 0. This allows us to finally show that the Steiner tree problem with edge lengths 1 and 2 cannot by approximated by polynomial time differential approximation schemata.
Approximation algorithms for spanner problems and Directed Steiner Forest
Information and Computation, 2013
We present an O(√ n log n)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonnegative edge lengths d : E → R ≥0 and a stretch k ≥ 1, a subgraph H = (V, E H) is a k-spanner of G if for every edge (s, t) ∈ E, the graph H contains a path from s to t of length at most k • d(s, t). The previous best approximation ratio wasÕ(n 2/3), due to Dinitz and Krauthgamer (STOC '11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths fromÕ(√ n) to O(n 1/3 log n). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer's lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n 1/3 log n)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(√ n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimumcost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n 2/3+) for any constant > 0, which improves the O(n • min(n 4/5 , m 2/3)) ratio due to Feldman, Kortsarz and Nutov (SODA '09).
An estimate of the objective function optimum for the network Steiner problem
Annals of Operations Research, 2015
A complete weighted graph, G(X, Γ, W), is considered. LetX ⊂ X be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes setX. The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of verticesX The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one.
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.
Kruskal-based approximation algorithm for the multi-level Steiner tree problem
2020
We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals TTT require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals uuu, vvv with priorities P(u)P(u)P(u), P(v)P(v)P(v) are connected using edges of rate minP(u),P(v)\min\{P(u),P(v)\}minP(u),P(v) or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio cloglognc \log \log ncloglogn, where nnn is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a min2(ln∣T∣+1),ellrho\min\{2(\ln |T|+1), \ell \rho\}min2(ln∣T∣+1),ellrho-approximation in this setting, where rho\rhorho is an approximation ratio for a heuristic solver for the Steiner tree problem and ell\ellell is the number of priorities or levels (Byrka et al. give a Steiner tree a...