New Approaches for Virtual Private Network Design

Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10, 2010

The Steiner tree problem is one of the most fundamental AEÈ-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from ¾ to the current best ½ [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than ¾ [Vazirani,Rajagopalan-SODA'99].

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.

Combinatorial Optimization, 2000

Many VLSI routing applications, as well as the facility location problem involve computation of Steiner trees with non-linear cost measures. We consider two most frequent versions of this problem. In the power-p Steiner problem the cost is de ned as the sum of the edge lengths where each length is raised to the power p > 1. In the bottleneck Steiner problem the objective cost is the maximum of the edge lengths. We show that the power-p Steiner problem is MAX SNP-hard and that one cannot guarantee to nd a bottleneck Steiner tree within a factor less than 2, unless P = NP. We prove that in any metric space the minimum spanning tree is at most a constant times worse than the optimal power-p Steiner tree. In particular, for p = 2, we show that the minimum spanning tree is at most 23.3 times worse than the optimum and we construct an instance for which it is 17.2 times worse. We also present a better approximation algorithm for the bottleneck Steiner problem with performance guarantee log 2 n, where n is the number of terminals (the minimum spanning tree can be 2 log 2 n times worse than the optimum).

Given two sets of points in the plane, $P$ of $n$ terminals and $S$ of $m$ Steiner points, a Steiner tree of $P$ is a tree spanning all points of $P$ and some (or none or all) points of $S$. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, $P$ and $S$, and a positive integer $k \le m$, find a bottleneck Steiner tree of $P$ with at most $k$ Steiner points. The problem has application in the design of wireless communication networks. We first show that the problem is NP-hard and cannot be approximated within factor $\sqrt{2}$, unless $P=NP$. Then, we present a polynomial-time approximation algorithm with performance ratio 2.

Siam Journal on Computing, 1995

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 {i, j} of nodes, an edge-connectivity requirement Tij. The goal is to find a minimum-cost network using the available links and satisfying the requirements.

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.

2019

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science. Recently, Grandoni, Laekhanukit and Li and independently Ghuge and Nagarajan gave quasi-polynomial time O(log^2k/loglog k)-approximation algorithms for the problem, which is tight under popular complexity assumptions. In this paper, we show a general framework that achieves O(log n log k)-approximation for many variants of DST. We show that if the problem has the property that the validity of the tree and its cost can be checked and computed using a bottom-to-top procedure, then the general framework can be used to produce a small-cost multi-tree: a tree that may contain multiple copies of a vertex or an edge. Using the framework, we show that two prominent variants of DST, namely Length-Bounded DST (LB-DST) and Buy-at-Bulk DST with Concave Cost Functions (BaB-DST-Concave), admit O(log n log k)-approximation algorithms. In the Length-Bounded Directed Steiner Tree (LB-DST)...

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.

Cornell University - arXiv, 2022

The k-Steiner-2NCS problem is as follows: Given a constant k, and an undirected connected graph G = (V, E), non-negative costs c on the edges, and a partition (T, V \ T) of V into a set of terminals, T , and a set of non-terminals (or, Steiner nodes), where |T | = k, find a min-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized FPTAS for the weighted problem. We obtain similar results for the k-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a min-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Björklund, Husfeldt, and Taslaman (SODA 2012) that give a randomized polynomial-time algorithm for the unweighted k-Steiner-cycle problem; this problem has the same inputs as the unweighted k-Steiner-2NCS problem, and the algorithmic goal is to find a min-cost simple cycle C that contains the terminals (C may contain any number of Steiner nodes).

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.