Approximation Algorithms for Network Design and Orienteering

This thesis presents approximation algorithms for some NP-Hard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widely-believed complexity-theoretic assumption that P is not equal to NP, there are no efficient (i.e., polynomial-time) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an NP-Hard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing fault-tolerant networks. Two vertices u,v in a network are said to be k-edge-connected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are k-vertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning that even if a small number of edges and nodes fail, the remaining nodes will still be able to communicate. A brief description of some of our results is given below. We study the problem of building 2-vertex-connected networks that are large and have low cost. Given an n-node graph with costs on its edges and any integer k, we give an O(log n log k) approximation for the problem of finding a minimum-cost 2-vertex-connected subgraph containing at least k nodes. We also give an algorithm of similar approximation ratio for maximizing the number of nodes in a 2-vertex-connected subgraph subject to a budget constraint on the total cost of its edges. Our algorithms are based on a pruning process that, given a 2-vertex-connected graph, finds a 2-vertex-connected subgraph of any desired size and of density comparable to the input graph, where the density of a graph is the ratio of its cost to the number of vertices it contains. This pruning algorithm is simple and efficient, and is likely to find additional applications. Recent breakthroughs on vertex-connectivity have made use of algorithms for element-connectivity problems. We develop an algorithm that, given a graph with some vertices marked as terminals, significantly simplifies the graph while preserving the pairwise element-connectivity of all terminals; in fact, the resulting graph is bipartite. We believe that our simplification/reduction algorithm will be a useful tool in many settings. We illustrate its applicability by giving algorithms to find many trees that each span a given terminal set, while being disjoint on edges and non-terminal vertices; such problems have applications in VLSI design and other areas. We also use this reduction algorithm to analyze simple algorithms for single-sink network design problems with high vertex-connectivity requirements; we give an O(k log n)-approximation for the problem of k-connecting a given set of terminals to a common sink. We study similar problems in which different types of links, of varying capacities and costs, can be used to connect nodes; assuming there are economies of scale, we give algorithms to construct low-cost networks with sufficient capacity or bandwidth to simultaneously support flow from each terminal to the common sink along many vertex-disjoint paths. We further investigate capacitated network design, where edges may have arbitrary costs and capacities. Given a connectivity requirement R_uv for each pair of vertices u,v, the goal is to find a low-cost network which, for each uv, can support a flow of R_uv units of traffic between u and v. We study several special cases of this problem, giving both algorithmic and hardness results. In addition to Network Design, we consider certain Traveling Salesperson-like problems, where the goal is to find short walks that visit many distinct vertices. We give a (2 + epsilon)-approximation for Orienteering in undirected graphs, achieving the best known approximation ratio, and the first approximation algorithm for Orienteering in directed graphs. We also give improved algorithms for Orienteering with time windows, in which vertices must be visited between specified release times and deadlines, and other related problems. These problems are motivated by applications in the fields of vehicle routing, delivery and transportation of goods, and robot path planning.

Preventing encrypted traffic analysis

Many existing encrypted Internet protocols leak information through packet sizes and timing. Though seemingly innocuous, prior work has shown that such leakage can be used to recover part or all of the plaintext being encrypted. The prevalence of encrypted protocols as the underpinning of such critical services as e-commerce, remote login, and anonymity networks and the increasing feasibility of attacks on these services represent a considerable risk to communications security. Existing mechanisms for preventing traffic analysis focus on re-routing and padding. These prevention techniques have considerable resource and overhead requirements. Furthermore, padding is easily detectable and, in some cases, can introduce its own vulnerabilities. To address these shortcomings, we propose embedding real traffic in synthetically generated encrypted cover traffic. Novel to our approach is our use of realistic network protocol behavior models to generate cover traffic. The observable traffic we generate also has the benefit of being indistinguishable from other real encrypted traffic further thwarting an adversary's ability to target attacks. In this dissertation, we introduce the design of a proxy system called TrafficMimic that implements realistic cover traffic tunneling and can be used alone or integrated with the Tor anonymity system. We describe the cover traffic generation process including the subtleties of implementing a secure traffic generator. We show that TrafficMimic cover traffic can fool a complex protocol classification attack with 91% of the accuracy of real traffic. TrafficMimic cover traffic is also not detected by a binary classification attack specifically designed to detect TrafficMimic. We evaluate the performance of tunneling with independent cover traffic models and find that they are comparable, and, in some cases, more efficient than generic constant-rate defenses. We then use simulation and analytic modeling to understand the performance of cover traffic tunneling more deeply. We find that we can take measurements from real or simulated traffic with no tunneling and use them to estimate parameters for an accurate analytic model of the performance impact of cover traffic tunneling. Once validated, we use this model to better understand how delay, bandwidth, tunnel slowdown, and stability affect cover traffic tunneling. Finally, we take the insights from our simulation study and develop several biasing techniques that we can use to match the cover traffic to the real traffic while simultaneously bounding external information leakage. We study these bias methods using simulation and evaluate their security using a Bayesian inference attack. We find that we can safely improve performance with biasing while preventing both traffic analysis and defense detection attacks. We then apply these biasing methods to the real TrafficMimic implementation and evaluate it on the Internet. We find that biasing can provide 3-5x improvement in bandwidth for bulk transfers and 2.5-9.5x speedup for Web browsing over tunneling without biasing.