Approximation Algorithms for Steiner Tree Problems Based on Universal Solution Frameworks

This paper summarizes the work on implementing few solutions for the Steiner Tree problem which we undertook in the PAAL project. The main focus of the project is the development of generic implementations of approximation algorithms together with universal solution frameworks. In particular, we have implemented Zelikovsky 11/6-approximation using local search framework, and 1.39-approximation by Byrka et al. using iterative rounding framework. These two algorithms are experimentally compared with greedy 2-approximation, with exact but exponential time Dreyfus-Wagner algorithm, as well as with results given by a state-of-the-art local search techniques by Uchoa and Werneck. The results of this paper are twofold. On one hand, we demonstrate that high level algorithmic concepts can be designed and efficiently used in C++. On the other hand, we show that the above algorithms with good theoretical guarantees, give decent results in practice, but are inferior to state-of-the-art heuristical approaches

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

Applicability of group communication for increased scalability in MMOGs

Massive multiplayer online games (MMOGs) are today the driving factor for the development of distributed interactive applications, and they are increasing in size and complex-ity. Even a small MMOG supports thousands of players, the biggest support hundreds of thousands of concurrent players. Since they are typically built as strict client-server systems, they suffer from the inherent scalability problem of the architecture. Computing power and bandwidth limita-tions close to the server limit the possible number of players. Also, the latency of communication between players through the server will be higher than using direct communication. In the paper, we address these issues and investigate im-provement options. A typical MMOG consists of a virtual world with a con-cept of time and space that is similar to the real world. In it, players are represented by avatars. Only subsets of these avatars interact with each other at any given time. This allows us to divide them into groups, and communication among group members becomes a multi-party communica-tion problem. Thus, to reduce resource consumption, we compare the performance of several algorithms for group communication with the current central server approach. We use overlay multicast as the means of providing group communication, and research algorithms for creating short-est path trees, spanning trees, delay-bounded spanning trees and, more specific, applying Steiner tree heuristics. Our experimental results indicate that different approaches are useful to reduce resource consumption while achieving a good perceived quality under varying conditions, such as frequent changes in group membership and the demand for low latency. 1

On the Implementation of MST-based Heuristics for the Steiner Problem in Graphs

Some of the most widely used constructive heuristics for the Steiner Problem in Graphs are based on algorithms for the Minimum Spanning Tree problem. In this paper, we examine efficient implementations of heuristics based on the classic algorithms by Prim, Kruskal, and Boruvka. An extensive experimental study indicates that the theoretical worst-case complexity of the algorithms give little information about their behavior in practice. Careful implementation improves average computation times not only significantly, but asymptotically. Running times for our implementations are within a small constant factor from that of Prim's algorithm for the Minimum Spanning Tree problem, suggesting that there is little room for improvement

Network Coding in Distributed, Dynamic, and Wireless Environments: Algorithms and Applications

The network coding is a new paradigm that has been shown to improve throughput, fault tolerance, and other quality of service parameters in communication networks. The basic idea of the network coding techniques is to relish the "mixing" nature of the information flows, i.e., many algebraic operations (e.g., addition, subtraction etc.) can be performed over the data packets. Whereas traditionally information flows are treated as physical commodities (e.g., cars) over which algebraic operations can not be performed. In this dissertation we answer some of the important open questions related to the network coding. Our work can be divided into four major parts. Firstly, we focus on network code design for the dynamic networks, i.e., the networks with frequently changing topologies and frequently changing sets of users. Examples of such dynamic networks are content distribution networks, peer-to-peer networks, and mobile wireless networks. A change in the network might result in infeasibility of the previously assigned feasible network code, i.e., all the users might not be able to receive their demands. The central problem in the design of a feasible network code is to assign local encoding coefficients for each pair of links in a way that allows every user to decode the required packets. We analyze the problem of maintaining the feasibility of a network code, and provide bounds on the number of modifications required under dynamic settings. We also present distributed algorithms for the network code design, and propose a new path-based assignment of encoding coefficients to construct a feasible network code. Secondly, we investigate the network coding problems in wireless networks. It has been shown that network coding techniques can significantly increase the overall throughput of wireless networks by taking advantage of their broadcast nature. In wireless networks each packet transmitted by a device is broadcasted within a certain area and can be overheard by the neighboring devices. When a device needs to transmit packets, it employs the Index Coding that uses the knowledge of what the device's neighbors have heard in order to reduce the number of transmissions. With the Index Coding, each transmitted packet can be a linear combination of the original packets. The Index Coding problem has been proven to be NP-hard, and NP-hard to approximate. We propose an efficient exact, and several heuristic solutions for the Index Coding problem. Noting that the Index Coding problem is NP-hard to approximate, we look at it from a novel perspective and define the Complementary Index Coding problem, where the objective is to maximize the number of transmissions that are saved by employing coding compared to the solution that does not involve coding. We prove that the Complementary Index Coding problem can be approximated in several cases of practical importance. We investigate both the multiple unicast and multiple multicast scenarios for the Complementary Index Coding problem for computational complexity, and provide polynomial time approximation algorithms. Thirdly, we consider the problem of accessing large data files stored at multiple locations across a content distribution, peer-to-peer, or massive storage network. Parts of the data can be stored in either original form, or encoded form at multiple network locations. Clients access the parts of the data through simultaneous downloads from several servers across the network. For each link used client has to pay some cost. A client might not be able to access a subset of servers simultaneously due to network restrictions e.g., congestion etc. Furthermore, a subset of the servers might contain correlated data, and accessing such a subset might not increase amount of information at the client. We present a novel efficient polynomial-time solution for this problem that leverages the matroid theory. Fourthly, we explore applications of the network coding for congestion mitigation and over flow avoidance in the global routing stage of Very Large Scale Integration (VLSI) physical design. Smaller and smarter devices have resulted in a significant increase in the density of on-chip components, which has given rise to congestion and over flow as critical issues in on-chip networks. We present novel techniques and algorithms for reducing congestion and minimizing over flows

Algorithmic Approaches to the Steiner Problem in Networks

Das Steinerproblem in Netzwerken ist das Problem, in einem gewichteten Graphen eine gegebene Menge von Knoten kostenminimal zu verbinden. Es ist ein klassisches NP-schweres Problem und ein fundamentales Problem bei der Netzwerkoptimierung mit vielen praktischen Anwendungen. Wir nehmen dieses Problem mit verschiedenen Mitteln in Angriff: Relaxationen, die die Zulässigkeitsbedingungen lockern, um eine optimale Lösung annähern zu können; Heuristiken, um gute, aber nicht garantiert optimale Lösungen zu finden; und Reduktionen, um die Probleminstanzen zu vereinfachen, ohne eine optimale Lösung zu zerstören. In allen Fällen untersuchen und verbessern wir bestehende Methoden, stellen neue vor und evaluieren sie experimentell. Wir integrieren diese Bausteine in einen exakten Algorithmus, der den Stand der Algorithmik für die optimale Lösung dieses Problems darstellt. Viele der vorgestellten Methoden können auch für verwandte Probleme von Nutzen sein