Improved Algorithms for the Steiner Problem in Networks

We present several new techniques for dealing with the Steiner problem in (undirected) networks. We consider them as building blocks of an exact algorithm, but each of them could also be of interest in its own right. First, we consider some relaxations of integer programming formulations of this problem and investigate different methods for dealing with these relaxations, not only to obtain lower bounds, but also to get additional information which is used in the computation of upper bounds and in reduction techniques. Then, we modify some known reduction tests and introduce some new ones. We integrate some of these tests into a package with a small worst case time which achieves impressive reductions on a wide range of instances. On the side of upper bounds, we introduce the new concept of heuristic reductions. On the basis of this concept, we develop heuristics that achieve sharper upper bounds than the strongest known heuristics for this problem despite running times which are smaller by orders of magnitude. Finally, we integrate these blocks into an exact algorithm. We present computational results on a variety of benchmark instances. The results are clearly superior to those of all other exact algorithms known to the authors

Further Improvements on Approximating the Uniform Cost-Distance Steiner Tree Problem

In this paper, we consider the Uniform Cost-Distance Steiner Tree Problem in metric spaces, a generalization of the well-known Steiner tree problem. Cost-distance Steiner trees minimize the sum of the total length and the weighted path lengths from a dedicated root to the other terminals, which have a weight to penalize the path length. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks, and the signal speed through the tree has to be considered besides the total length. Constant factor approximation algorithms for the uniform cost-distance Steiner tree problem have been known since the first mentioning of the problem by Meyerson, Munagala, and Plotkin. Recently, the approximation factor was improved from 2.87 to 2.39 by Khazraei and Held. We refine their approach further and reduce the approximation factor down to 2.15

Optimal competitiveness for Symmetric Rectilinear Steiner Arborescence and related problems

We present optimal competitive algorithms for two interrelated known problems involving Steiner Arborescence. One is the continuous problem of the Symmetric Rectilinear Steiner Arborescence (SRSA), studied by Berman and Coulston. A very related, but discrete problem (studied separately in the past) is the online Multimedia Content Delivery (MCD) problem on line networks, presented originally by Papadimitriu, Ramanathan, and Rangan. An efficient content delivery was modeled as a low cost Steiner arborescence in a grid of network*time they defined. We study here the version studied by Charikar, Halperin, and Motwani (who used the same problem definitions, but removed some constraints on the inputs). The bounds on the competitive ratios introduced separately in the above papers are similar for the two problems: O(log N) for the continuous problem and O(log n) for the network problem, where N was the number of terminals to serve, and n was the size of the network. The lower bounds were Omega(sqrt{log N}) and Omega(sqrt{log n}) correspondingly. Berman and Coulston conjectured that both the upper bound and the lower bound could be improved. We disprove this conjecture and close these quadratic gaps for both problems. We first present an O(sqrt{log n}) deterministic competitive algorithm for MCD on the line, matching the lower bound. We then translate this algorithm to become a competitive optimal algorithm O(sqrt{log N}) for SRSA. Finally, we translate the latter back to solve MCD problem, this time competitive optimally even in the case that the number of requests is small (that is, O(min{sqrt{log n},sqrt{log N}})). We also present a Omega(sqrt[3]{log n}) lower bound on the competitiveness of any randomized algorithm. Some of the techniques may be useful in other contexts

The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks

It is challenging to design large and low-cost communication networks. In this paper, we formulate this challenge as the prize-collecting Steiner Tree Problem (PCSTP). The objective is to minimize the costs of transmission routes and the disconnected monetary or informational profits. Initially, we note that the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to improve suboptimal solutions to PCSTP. Based on these techniques, we propose two fast heuristic algorithms: the first one is a quasilinear time heuristic algorithm that is faster and consumes less memory than other algorithms; and the second one is an improvement of a stateof-the-art polynomial time heuristic algorithm that can find high-quality solutions at a speed that is only inferior to the first one. We demonstrate the competitiveness of our heuristic algorithms by comparing them with the state-of-the-art ones on the largest existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we generate new instances that are even larger (1 000 000 vertices and 10 000 000 edges) to further demonstrate their advantages in large networks. The state-ofthe-art algorithms are too slow to find high-quality solutions for instances of this size, whereas our new heuristic algorithms can do this in around 6 to 45s on a personal computer. Ultimately, we apply our post-processing techniques to update the bestknown solution for a notoriously difficult benchmark instance to show that they can improve near-optimal solutions to PCSTP. In conclusion, we demonstrate the usefulness of our heuristic algorithms and post-processing techniques for designing large and low-cost communication networks

Demand-Aware Network Design with Steiner Nodes and a Connection to Virtual Network Embedding

Emerging optical and virtualization technologies enable the design of more flexible and demand-aware networked systems, in which resources can be optimized toward the actual workload they serve. For example, in a demand-aware datacenter network, frequently communicating nodes (e.g., two virtual machines or a pair of racks in a datacenter) can be placed topologically closer, reducing communication costs and hence improving the overall network performance. This paper revisits the bounded-degree network design problem underlying such demand-aware networks. Namely, given a distribution over communicating server pairs, we want to design a network with bounded maximum degree that minimizes expected communication distance. In addition to this known problem, we introduce and study a variant where we allow Steiner nodes (i.e., additional routers) to be added to augment the network. We improve the understanding of this problem domain in several ways. First, we shed light on the complexity and hardness of the aforementioned problems, and study a connection between them and the virtual networking embedding problem. We then provide a constant-factor approximation algorithm for the Steiner node version of the problem, and use it to improve over prior state-of-the-art algorithms for the original version of the problem with sparse communication distributions. Finally, we investigate various heuristic approaches to bounded-degree network design problem, in particular providing a reliable heuristic algorithm with good experimental performance. We report on an extensive empirical evaluation, using several real-world traffic traces from datacenters, and find that our approach results in improved demand-aware network designs