The set cover problem is a well studied problem in computer science. The problem cannot be approximated to better than an log n-factor in polynomial time unless P = NP and has an O(log n)-factor approximation algorithm. We consider several special cases of the set cover problem in which geometry plays a key role. With geometric structure introduced to the problem, it may be possible to construct approximation algorithms with approximation ratios asymptotically better than log n. The first problem we consider is the decomposing coverings problem. Here, we consider a combinatorial problem: given a collection of points in the plane and a collection of objects in the plane such that each point is contained in at least k objects, partition the objects into as many sets as possible so that each set covers all of the points. We show that if the objects are translates of a convex polygon, then it is possible to partition the translates into Ω(k) covers. The second problem we consider is the planar sensor cover problem. This problem is a generalization of the decomposing coverings problem. We are given a collection of points in the plane and a collection of objects in the plane. Each of the objects can be thought of as a sensor. The sensors have a duration which can be thought of as the battery life of the sensor. The planar sensor cover problem is to schedule a start time to each of the sensors so that the points are covered by a sensor for as long as possible. We give a constant factor approximation for this problem. The key contribution to this result is a constant factor approximation to a one-dimensional version of the problem called the restricted strip cover (RSC) problem. Our result for RSC improves upon the previous best O(log log log n)-approximation, and our result for the planar sensor cover problem improves upon the previous best O(log n)-approximation. The next problem we consider is the metric clustering to minimize the sum of radii problem. Here, we are given an n-point metric (P,d) and an integer k > 0. We are interested in covering the points in P with at most k balls so that the sum of the radii of the balls is minimized. We give a randomized algorithm which solves the problem exactly in nO(log n log Δ) time, where Δ is the ratio of the maximum interpoint distance to the minimum interpoint distance. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and when the metric has constant doubling dimension. The last problem we consider is the minimum dominating set problem for disk graphs. In this problem, we are given a set of disks in the plane, and we want to choose a minimum-cardinality subset of disks such that every disk is either in the set or intersects a disk in the set. For any ε > 0, we show that a simple local search algorithm is a (1+ ε)-approximation for the problem which improves upon the previous best O(log n)-approximation algorithm.

We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, one can prove that an optimal flow and an equilibrium flow share a desirable property in this situation: all flow-carrying paths have the same length; i.e., these solutions are "fair," which is in general not true for the optimal flow in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances that shows that the price of anarchy is unbounded for instances with multiple sources and a single sink, even in networks with linear latencies. Finally, we show that an s-t-flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latenc

The online and data stream models of computation have recently attracted considerable research attention due to many real-world applications in various areas such as data mining, machine learning, distributed computing, and robotics. In both these models, input items arrive one at a time, and the algorithms must decide based on the partial data received so far, without any secure information about the data that will arrive in the future. In this thesis, we investigate efficient algorithms for a number of fundamental geometric optimization problems in the online and data stream models. The problems studied in this thesis can be divided into two major categories: geometric clustering and computing various extent measures of a set of points. In the online setting, we show that the basic unit clustering problem admits non-trivial algorithms even in the simplest one-dimensional case: we show that the naive upper bounds on the competitive ratio of algorithms for this problem can be beaten using randomization. In the data stream model, we propose a new streaming algorithm for maintaining "core-sets" of a set of points in fixed dimensions, and also, introduce a new simple framework for transforming a class of offline algorithms to their equivalents in the data stream model. These results together lead to improved streaming approximation algorithms for a wide variety of geometric optimization problems in fixed dimensions, including diameter, width, k-center, smallest enclosing ball, minimum-volume bounding box, minimum enclosing cylinder, minimum-width enclosing spherical shell/annulus, etc. In high-dimensional data streams, where the dimension is not a constant, we propose a simple streaming algorithm for the minimum enclosing ball (the 1-center) problem with an improved approximation factor.

Modern routers use specialized hardware, such as Ternary Content Addressable Memory (TCAM), to solve the Longest Prefix Matching Problem (LPMP) quickly. Due to the fact that TCAM is a non-standard type of memory and inherently parallel, there are concerns about its cost and power consumption. This problem is exacerbated by the growth in routing tables, which demands ever larger TCAMs. To reduce the size of the TCAMs in a distributed forwarding environment, a batch caching model is proposed and analyzed. The problem of determining which routing prefixes to store in the TCAMs reduces to the In-tree Knapsack Problem (ITKP) for unit weight vertices in this model. Several algorithms are analysed for solving the ITKP, both in the general case and when the problem is restricted to unit weight vertices. Additionally, a variant problem is proposed and analyzed, which exploits the caching model to provide better solutions. This thesis concludes with discussion of open problems and future experimental work.

The online and data stream models of computation have recently attracted considerable research attention due to many real-world applications in various areas such as data mining, machine learning, distributed computing, and robotics. In both these models, input items arrive one at a time, and the algorithms must decide based on the partial data received so far, without any secure information about the data that will arrive in the future. In this thesis, we investigate efficient algorithms for a number of fundamental geometric optimization problems in the online and data stream models. The problems studied in this thesis can be divided into two major categories: geometric clustering and computing various extent measures of a set of points. In the online setting, we show that the basic unit clustering problem admits non-trivial algorithms even in the simplest one-dimensional case: we show that the naive upper bounds on the competitive ratio of algorithms for this problem can be beaten using randomization. In the data stream model, we propose a new streaming algorithm for maintaining "core-sets" of a set of points in fixed dimensions, and also, introduce a new simple framework for transforming a class of offline algorithms to their equivalents in the data stream model. These results together lead to improved streaming approximation algorithms for a wide variety of geometric optimization problems in fixed dimensions, including diameter, width, k-center, smallest enclosing ball, minimum-volume bounding box, minimum enclosing cylinder, minimum-width enclosing spherical shell/annulus, etc. In high-dimensional data streams, where the dimension is not a constant, we propose a simple streaming algorithm for the minimum enclosing ball (the 1-center) problem with an improved approximation factor.

Modern routers use specialized hardware, such as Ternary Content Addressable Memory (TCAM), to solve the Longest Prefix Matching Problem (LPMP) quickly. Due to the fact that TCAM is a non-standard type of memory and inherently parallel, there are concerns about its cost and power consumption. This problem is exacerbated by the growth in routing tables, which demands ever larger TCAMs. To reduce the size of the TCAMs in a distributed forwarding environment, a batch caching model is proposed and analyzed. The problem of determining which routing prefixes to store in the TCAMs reduces to the In-tree Knapsack Problem (ITKP) for unit weight vertices in this model. Several algorithms are analysed for solving the ITKP, both in the general case and when the problem is restricted to unit weight vertices. Additionally, a variant problem is proposed and analyzed, which exploits the caching model to provide better solutions. This thesis concludes with discussion of open problems and future experimental work.

Iterative rounding has been an increasingly popular approach to solving network design optimization problems ever since Jain introduced the concept in his revolutionary 2-approximation for the Survivable Network Design Problem (SNDP). This paper looks at several important iterative rounding approximation algorithms and makes improvements to some of their proofs. We generalize a matrix restatement of Nagarajan et al.'s token argument, which we can use to simplify the proofs of Jain's 2-approximation for SNDP and Fleischer et al.'s 2-approximation for the Element Connectivity (ELC) problem. Lau et al. show how one can construct a (2,2B + 3)-approximation for the degree bounded ELC problem, and this thesis provides the proof. We provide some structural results for basic feasible solutions of the Prize-Collecting Steiner Tree problem, and introduce a new problem that arises, which we call the Prize-Collecting Generalized Steiner Tree problem.

The subject of this thesis is to study approximation algorithms for some network design problems in face of uncertainty. We consider two widely studied models of handling uncertainties - Robust Optimization and Stochastic Optimization. We study a robust version of the well studied Uncapacitated Facility Location Problem (UFLP). In this version, once the set of facilities to be opened is decided, an adversary may close at most β facilities. The clients must then be assigned to the remaining open facilities. The performance of a solution is measured by the worst possible set of facilities that the adversary may close. We introduce a novel LP for the problem, and provide an LP rounding algorithm when all facilities have same opening costs. We also study the 2-stage Stochastic version of the Steiner Tree Problem. In this version, the set of terminals to be covered is not known in advance. Instead, a probability distribution over the possible sets of terminals is known. One is allowed to build a partial solution in the first stage a low cost, and when the exact scenario to be covered becomes known in the second stage, one is allowed to extend the solution by building a recourse network, albeit at higher cost. The aim is to construct a solution of low cost in expectation. We provide an LP rounding algorithm for this problem that beats the current best known LP rounding based approximation algorithm.

We consider several related problems arising in geometric graphs. In particular, we investigate the computational complexity and approximability properties of several optimization problems in unit ball graphs and develop algorithms to find exact and approximate solutions. In addition, we establish complexity-based theoretical justifications for several greedy heuristics. Unit ball graphs, which are defined in the three dimensional Euclidian space, have several application areas such as computational geometry, facility location and, particularly, wireless communication networks. Efficient operation of wireless networks involves several decision problems that can be reduced to well known optimization problems in graph theory. For instance, the notion of a \virtual backbone" in a wire- less network is strongly related to a minimum connected dominating set in its graph theoretic representation. Motivated by the vastness of application areas, we study several problems including maximum independent set, minimum vertex coloring, minimum clique partition, max-cut and min-bisection. Although these problems have been widely studied in the context of unit disk graphs, which are the two dimensional version of unit ball graphs, there is no established result on the complexity and approximation status for some of them in unit ball graphs. Furthermore, unit ball graphs can provide a better representation of real networks since the nodes are deployed in the three dimensional space. We prove complexity results and propose solution procedures for several problems using geometrical properties of these graphs. We outline a matching-based branch and bound solution procedure for the maximum k-clique problem in unit disk graphs and demonstrate its effectiveness through computational tests. We propose using minimum bottleneck connected dominating set problem in order to determine the optimal transmission range of a wireless network that will ensure a certain size of "virtual backbone". We prove that this problem is NP-hard in general graphs but solvable in polynomial time in unit disk and unit ball graphs. We also demonstrate work on theoretical foundations for simple greedy heuristics. Particularly, similar to the notion of "best" approximation algorithms with respect to their approximation ratios, we prove that several simple greedy heuristics are "best" in the sense that it is NP-hard to recognize the gap between the greedy solution and the optimal solution. We show results for several well known problems such as maximum clique, maximum independent set, minimum vertex coloring and discuss extensions of these results to a more general class of problems. In addition, we propose a "worst-out" heuristic based on edge contractions for the max-cut problem and provide analytical and experimental comparisons with a well known "best-in" approach and its modified versions.

Thesis (Ph. D.)--Carnegie Mellon University, 2003. / Title from PDF title page (viewed Dec. 18, 2009). Includes bibliographical references (p. 49-52).