Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.

Approximation Algorithms

Many DTN routing protocols use a variety of mechanisms, including discovering the meeting probabilities among nodes, packet replication, and network coding. The primary focus of these mechanisms is to increase the likelihood of finding a path with limited information, so these approaches have only an incidental effect on such routing metrics as maximum or average delivery latency. In this paper, we present RAPID, an intentional DTN routing protocol that can optimize a specific routing metric such as worst-case delivery latency or the fraction of packets that are delivered within a deadline. The key insight is to treat DTN routing as a resource allocation problem that translates the routing metric into per-packet utilities which determine how packets should be replicated in the system.We evaluate RAPID rigorously through a prototype of RAPID deployed over a vehicular DTN testbed of 40 buses and simulations based on real traces. To our knowledge, this is the first paper to report on a routing protocol deployed on a real DTN at this scale. Our results suggest that RAPID significantly outperforms existing routing protocols for several metrics. We also show empirically that for small loads RAPID is within 10% of the optimal performance.

/pdf/dtn-routing-as-a-resource-allocation-problem-22qq5k3jz5.pdf

DTN routing as a resource allocation problem

Discrete optimization problems are everywhere, from traditional operations research planning problems, such as scheduling, facility location, and network design; to computer science problems in databases; to advertising issues in viral marketing. Yet most such problems are NP-hard. Thus unless P = NP, there are no efficient algorithms to find optimal solutions to such problems. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first part of the book is devoted to a single algorithmic technique, which is then applied to several different problems. The second part revisits the techniques but offers more sophisticated treatments of them. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithms courses, the book will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

https://www.designofapproxalgs.com/book.pdf

The Design of Approximation Algorithms

We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that /spl epsiv/-approximates the Pareto curve, for any /spl epsiv/>0. We give a necessary and sufficient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1//spl epsiv/. In the case of multiple linear objectives, we distinguish between two cases: when the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If however the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: how does the complexity of a combinatorial optimization problem change when its feasible region is intersected with a hyperplane with small coefficients; we report some interesting new findings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the World-Wide Web, in a model first studied by Etzioni et al. (1996) (which was actually the original motivation for this work).

On the approximability of trade-offs and optimal access of Web sources

Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints "communicated." We will call such a graph a temporal network. To model the notion that information in such a network "flows" only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. We provide results on two types of problems for temporal networks. First, we consider connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes. The natural analogue of Menger's Theorem for node-disjoint paths fails in general for time-respecting paths; we give a non-trivial characterization of those graphs for which the theorem does hold in terms of an excluded subdivision theorem, and provide a polynomial-time algorithm for connectivity on this class of graphs. (The problem on general graphs is NP-complete.) We then define and study the class of inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow.

Connectivity and inference problems for temporal networks

We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si; ti), 1 i k, and the goal is to nd a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any > 0, EDP is NP-hard to approximate within m 1=2 . We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any > 0, bounded length EDP is hard to approximate within m 1=2 even in undirected networks, and give an O( p m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. nd the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1=2 , for any > 0.

/pdf/near-optimal-hardness-results-and-approximation-algorithms-5dau4hnlub.pdf

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

We consider the problem of client assignment in a distributed system of content servers. We present a system called Webmapper for clustering IP addresses and assigning each cluster to an optimal content server. The system is passive in that the only information it uses comes from monitoring the TCP connections between the clients and the servers. It is also flexible in that it makes no a priori assumptions about network topology and server placement and it can react quickly to changing network conditions. We present experimental results to evaluate the performance of Webmapper.

Clustering and server selection using passive monitoring

In the d-dimensional bin packing problem (VBP), one is given vectors x1,x2,  ,xn ∈ Rd and the goal is to find a partition into a minimum number of feasible sets: {1,2  ,n} = ∪is Bi A set Bi is feasible if ∑j ∈ Bi xj ≤ 1, where 1 denotes the all 1's vector For online VBP, it has been outstanding for almost 20 years to clarify the gap between the best lower bound Ω(1) on the competitive ratio versus the best upper bound of O(d) We settle this by describing a Ω(d1-e) lower bound We also give strong lower bounds (of Ω(d1/B-e) ) if the bin size B ∈ Z+ is allowed to grow Finally, we discuss almost-matching upper bound results for general values of B; we show an upper bound whose exponent is additively "shifted by 1" from the lower bound exponent

Tight bounds for online vector bin packing

We introduce a game theoretic model of network formation in an effort to understand the complex system of business relationships between various Internet entities (e.g., Autonomous Systems, enterprise networks, residential customers). This system is at the heart of Internet connectivity. In our model we are given a network topology of nodes and links where the nodes (modeling the various Internet entities) act as the players of the game, and links represent potential contracts. Nodes wish to satisfy their demands, which earn potential revenues, but nodes may have to pay (or be paid by) their neighbors for links incident to them. By incorporating some of the qualities of Internet business relationships, we hope that our model will have predictive value. Specifically, we assume that contracts are either customer-provider or peering contracts. As often occurs in practice, we also include a mechanism that penalizes nodes if they drop traffic emanating from one of their customers. For a natural objective function, we prove that the price of stability is at most 2. With respect to social welfare, however, the prices of anarchy and stability can both be unbounded, leading us to consider how much we must perturb the system to obtain good stable solutions. We thus focus on the quality of Nash equilibria achievable through centralized incentives: solutions created by an "altruistic entity" (e.g., the government) able to increase individual payouts for successfully routing a particular demand. We show that if every payout is increased by a factor of 2, then there is a Nash equilibrium as good as the original centrally defined social optimum. We also show how to find equilibria efficiently in multicast trees. Finally, we give a characterization of Nash equilibria as flows of utility with certain constraints, which helps to visualize the structure of stable solutions and provides us with useful proof techniques.

/pdf/strategic-network-formation-through-peering-and-service-3i1p957egt.pdf

Bruce Shepherd

Papers

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

Clustering and server selection using passive monitoring

Tight bounds for online vector bin packing

Strategic Network Formation through Peering and Service Agreements