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).

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Στην παρούσα διδακτορική διατριβή ασχολούμαστε με ζητήματα ελαχιστοποίησης της κατανάλωσης ενέργειας που ανακύπτουν σε ασύρματα δίκτυα. Εξετάζουμε τόσο την περίπτωση ασυρμάτων δικτύων τύπου ad hoc όσο και την περίπτωση όπου υπάρχει ένα σταθερό ενσύρματο δίκτυο το οποίο συνδέει τους σταθμούς εκπομπής, οι οποίοι χρησιμοποιούν ασύρματα μέσα προκειμένου να μεταδώσουν μηνύματα στους χρήστες. Στην πρώτη κατηγορία, μελετούμε τόσο περιπτώσεις όπου η συνάρτηση κόστους στις ακμές είναι συμμετρική, όσο και περιπτώσεις όπου δεν ισχύει αυτή η υπόθεση. Εξετάζουμε επιπλέον προβλήματα που προκύπτουν όταν θεωρούμε ότι οι σταθμοί βρίσκονται σε κάποιον Ευκλείδειο χώρο και η απόσταση εξαρτάται από την Ευκλείδεια απόσταση. Παρουσιάζουμε αποτελέσματα υπολογιστικής δυσκολίας για την εύρεση τόσο της βέλτιστης λύσης όσο και μιας καλής προσεγγιστικής λύσης. Από την άλλη πλευρά, αποδεικνύουμε άνω φράγματα στον λόγο προσέγγισης διάφορων πολυωνυμικών αλγορίθμων. Στην περίπτωση που θεωρούμε πως οι σταθμοί μετάδοσης είναι συνδεδεμένοι με ένα ενσύρματο δίκτυο, έχουμε το πρόβλημα της συσταδοποίησης. Παρουσιάζουμε έναν βέλτιστο πολυωνυμικό αλγόριθμο για την περίπτωση όπου τα σημεία είναι συνευθειακά, ενώ αποδεικνύουμε αποτελέσματα υπολογιστικής δυσκολίας για την περίπτωση των δύο ή περισσοτέρων διαστάσεων. Τέλος, παρουσιάζουμε έναν προσεγγιστικό αλγόριθμο του οποίου ο λόγος προσέγγισης μπορεί να πλησιάσει αυθαίρετα κόντα το 1, με άλλα λόγια παρουσιάζουμε ένα προσεγγιστικό σχήμα πολυωνυμικού χρόνου. / In this dissertation we focus on issues related to energy consumption in wireless networks. We examine both ad hoc wireless networks, where we assume that there is no wired infrastructure, and networks where antennas are wired through a traditional, wired backbone network but they transmit messages to the users using wireless means. In the first case, we consider networks where the distance function can be symmetric or asymmetric; asymmetric edge cost functions can be used to model medium abnormalities or batteries with different energy levels. We prove results concerning the NP-hardness of computing the optimal solution or in some cases even an approximate solution, and also present upper bounds on the approximation ratio of several polynomial time algorithms. In the case where the antennas are connected through a wired backbone network, we consider a clustering problem. We present an optimal polynomial time algorithm for the special case when points are located on a line. We also present NP-hardness results concerning special cases of the problem in the case of 2 or more dimensions. Finally, we conclude with a polynomial time approximation scheme (PTAS).