Definition 9.1: A graph G is said to be connected if for every pair of vertices there is a path joining them. The maximal connected subgraphs are called components. Definition 9.2: The connectivity number κ(G) is defined as the minimum number of vertices whose removal from G results in a disconnected graph or in the trivial graph (=a single vertex). A graph G is said to be k-connected if κ(G) ≥ k. Definition 9.3: The connectivity number λ(G) is defined as the minimum number of edges whose removal from G results in a disconnected graph or in the trivial graph (=a single vertex). A graph G is said to be k-edge-connected if λ(G) ≥ k. Proof: Let v be a vertex with d(v) = δ(G), then removing all edges incident to v disconnects v from the other vertices of G. Therefore, λ(G) ≤ δ(G). If λ(G) = 0 or 1, then κ(G) = λ(G). On the other hand, if λ(G) = k ≥ 2, let x 1 y 1 , x 2 y 2 ,. .. , x k y k are be the edges whose removal causes G to be disconnected (where some of the x i , resp. y i , vertices might be identical). Denote V 1 and V 2 as the components of this disconnected graph.

Dedicated to my mother, who probably won't understand a word of this... All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. ACKNOWLEDGMENTS First of all, I would like to thank God for His continuous grace and mercy over my life. Next, I would like to thank Dr. Mittal for his support as my advisor. He has given me the opportunity to perform this research, and has served as a helpful guide concerning viable research directions. I must also acknowledge Dr. Raghavachari's insight concerning the method for visiting subproblems, and for many long, fruitful discussions. I would also like to thank Dr. Chandra and Dr. Venky for their helpful comments and especially for their patience concerning frequent visits outside of their regular office hours. We would also like to thank Dr. Shashidhar Gandham, a Ph.D graduate from UT Dallas currently working at Google, for his helpful suggestions concerning applications of our failure model to network failures. I cannot close without mentioning the support I have received from the Erik Jonsson School of Engineering and Computer Science by way of a fellowship which has given me the opportunity to perform this research. Finally, I wish to thank my family. Without their support, I would not have been able to come this far. Thank you all. April 2012 v PREFACE This thesis was produced in accordance with guidelines which permit the inclusion as part of the thesis the text of an original paper or papers submitted for publication. The thesis must still conform to all other requirements explained in the " Guide for the Preparation of Master's Theses and Doctoral Dissertations at The University of Texas at Dallas. " It must include a comprehensive abstract, a full introduction and literature review, and a final overall conclusion. Additional material (procedural and design data as well as descriptions of equipment) must be provided in sufficient detail to allow a clear and precise judgment to be made of the importance and originality of the research reported. It is acceptable for this thesis to include as chapters authentic copies of papers already published, provided these meet type size, margin, and …

DOMINE is a comprehensive collection of known and predicted domain-domain interactions (DDIs) compiled from 15 different sources. The updated DOMINE includes 2285 new domain-domain interactions (DDIs) inferred from experimentally characterized high-resolution three-dimensional structures, and about 3500 novel predictions by five computational approaches published over the last 3 years. These additions bring the total number of unique DDIs in the updated version to 26 219 among 5140 unique Pfam domains, a 23% increase compared to 20 513 unique DDIs among 4346 unique domains in the previous version. The updated version now contains 6634 known DDIs, and features a new classification scheme to assign confidence levels to predicted DDIs. DOMINE will serve as a valuable resource to those studying protein and domain interactions. Most importantly, DOMINE will not only serve as an excellent reference to bench scientists testing for new interactions but also to bioinformaticans seeking to predict novel protein-protein interactions based on the DDIs. The contents of the DOMINE are available at http://domine.utdallas.edu.

In this work, we study the NP-hard problem of finding the path of maximum reliability between two end nodes in mesh networks against simultaneous failures of multiple links. The links belong to shared risk link groups (SRLGs) that have arbitrary failure probabilities. We propose a novel algorithm to find the optimal path with semi-polynomial running time and prove its correctness.

Edge (u; v) is incident to u and v Degree of vertex in undirected graph is the number of edges incident to it. In (Out) degree of a vertex in directed graph is the number of edges entering (leaving) it. A path from u1 to u2 is a sequence of vertices < u1=v0; v1; v2; ; vk=u2 > such that (vi; vi+1) 2 E (or fvi; vi+1g 2 E) { We say that u2 is reachable from u1 { The length of the path is k { It is a cycle if v0 = vk

In this work, we study the problems of finding the path of maximum survivability between two end nodes. Multiple network devices and connections may fail simultaneously due to either deliberate attacks or components failures. The problem is NP-hard and we propose a novel algorithm to find the optimal paths with semi-polynomial running time and prove its correctness.

Given n points in the Euclidean plane, the degree-δ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most δ. The problem is NP-hard for 2≤δ≤3, while the NP-hardness of the problem is open for δ=4. The problem is polynomial-time solvable when δ=5. By presenting an improved approximation analysis for Chan’s degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177–194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most (2+2)/3<1.1381 times the weight of an MST.

Given an undirected graph G = (V, E) and a source vertex s is an element of V, the k-traveling repairman (KTR) problem, also known as the minimum latency problem, asks for k tours, each starting at s and together covering all the vertices (customers) such that the sum of the latencies experienced by the customers is minimum. The latency of a customer p is defined to be the distance traveled (time elapsed) before visiting p for the first time. Previous literature on the KTR problem has considered the version of the problem in which the repairtime of a customer is assumed to be zero for latency calculations. We consider a generalization of the problem in which each customer has an associated repairtime. For a fixed k, we present a (beta + 2)-approximation algorithm for this problem, where beta is the best achievable approximation ratio for the KTR problem with zero repairtimes (currently beta = 6). For arbitrary k, we obtain a (3/2 beta + 1/2)-approximation ratio. When the repairtimes of the customers are all the same, we present an approximation algorithm with a better ratio.(2) We also introduce the bounded-latency problem, a complementary version of the KTR problem, in which we are given a latency bound L and are asked to find the minimum number of repairmen required to service all the customers such that the latency of no customer is more than L. For this problem, we present a simple bicriteria approximation algorithm that finds a solution with at most 2/rho times the number of repairmen required by an optimal solution, with the latency of no customer exceeding (1 + rho) L, rho > 0. (C) 2006 Elsevier B. V. All rights reserved.

DOMINE is a database of known and predicted protein domain interactions compiled from a variety of sources. The database contains domain-domain interactions observed in PDB entries, and those that were predicted by eight different computational approaches. DOMINE contains a total of 20 513 unique domain-domain interactions among 4036 Pfam domains, out of which 4349 are inferred from PDB entries and 17 781 were predicted by at least one computational approach. This database will serve as a valuable resource to those working in the field of protein and domain interactions. DOMINE may not only serve as a reference to experimentalists who test for new protein and domain interactions, but also offers a consolidated dataset for analysis by bioinformaticians who seek to test ideas regarding the underlying factors that control the topological structure of interaction networks. DOMINE is freely available at http://domine.utdallas.edu.

The minimum cardinality 3-edge-connected spanning subgraph problem is considered. An approximation algorithm with a performance ratio of 4/3 ≈ 1.33 is presented. This improves the previous best ratio of 3/2 for the problem. The algorithm also works on multigraphs and guarantees the same approximation ratio.

We consider optimization problems where the main constraint is connectivity. Finding minimum-cost subgraphs with connectivity requirements is a fundamental problem in network optimization. This dissertation focuses on two important problems: the 2-vertex-connectivity problem and the 3-edge-connectivity problem. Improving approximation ratios for k-connectivity problems when k is small is quite a challenge because the problems are easily understood and fundamental, and a lot of researchers have worked on them and many papers have already been published. So, we had to come up with nontrivial algorithms to beat current ratios. Also, one of the uses of approximation algorithms for the above problems is that they can be used as black box algorithms for solving higher connectivity problems. Given a 2-connected graph G = (V, E), the 2-vertex-connected-spanning-subgraph (2VCSS) problem is to find a minimum cardinality subgraph that is also 2-connected. Given a 3-edge-connected graph G(V, E), the 3-edge-connected-spanning-subgraph (3ECSS) problem is to find a minimum cardinality subgraph that is also 3-edge-connected. We also discuss the average- k-VCSS problem and average-k-ECSS problem in this thesis. These problems deal with finding k-connected spanning subgraphs of minimum average weight. We present different approximation algorithms for edge connectivity and vertex connectivity problems. The following results are discussed in the dissertation: (1) An approximation algorithm with a performance ratio of 9/7 &ap; 1.286 for the 2VCSS problem. (2) An approximation algorithm with a performance ratio of 5/4 for the 2VCSS problem in graphs with a minimum-degree of three at every node. (3) An approximation algorithm with a performance ratio of 4/3 &ap; 1.33 is presented for the 3ECSS problem. (4) A 3-approximation algorithm for the average-k-ECSS problem. (5) An O(logk) approximation algorithm for the average- k-VCSS problem. (6) A 2 + &epsi; approximation algorithm for the average-k-VCSS in Euclidean graphs, for any constant &epsi; 0. (7) A 3-approximation algorithm for the average- k-VCSS in graphs satisfying the triangle inequality.

We give quasipolynomial-time approximation algorithms for designing networks with minimum degree. Using our methods, one can design one-connected networks to meet a variety of connectivity requirements. The degree of the output network is guaranteed to be at most (1+ε) times optimal, plus an additive error of O(log n/ε) for any εs0. We also provide a quasipolynomial-time approximation algorithm for designing a two-edge-connected spanning subgraph of a given two-edge-connected graph of approximately minimum degree. The performance guarantee is identical to that for one-connected networks. As a consequence of our analysis, we show that the minimum degree in both the problems above is well-estimated by certain polynomially solvable linear programs. This fact suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch-andbound.

The minimum cardinality 2-connected spanning subgraph problem is considered. An approximation algorithm with a performance ratio of 9/7 ≈ 1.286 is presented. This improves the previous best ratio of 4/3 for the problem. An approximation algorithm with a performance ratio of 5/4 in graphs with a minimum-degree of three at every node is also presented. This class includes all 3-connected graphs.

The traditional multi-commodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multi-commodity flow network-design problem: given a set of multi-commodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimum-weight balanced-separator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced-separator procedure. If Leighton and Rao's balanced-separator procedure is used, the performance guarantee is O(log n).

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low-weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning treeTusingadoptionsto meet the degree constraints is considered. A novel network-flow-based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraintd(v) for eachvis at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 − min{(d(v) − 2)/(degT(v) − 2) : degT(v) > 2}, where degT(v) is the initial degree ofv. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds foranyalgorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. ChoosingTto be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by variousLpnorms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.

A fault-tolerant scheme, called dual homing, is generally used in IP-based access networks to increase the availability of the networks. In a dual-homing architecture, a host is connected to two different access routers; therefore, it is unlikely that the host will be denied access to the network as the result of an access line break, a defective power supply in the access router, or congestion of the access router. This dual-homing architecture in the access network imposes the overhead to provide protection in the core network. Scaling the next-generation IP-over-wavelength-division-multiplexing (WDM) Internet, and being able to support a growing number of such dual-homing connections, as well as protection, demands a scalable mechanism to contain this overhead for protection in the WDM networks. This paper studies the coordinated protection design to reduce the protection cost in the WDM core network, given a dual-homing infrastructure in the access network. The protection problem is considered for both static and dynamic traffic. Several solutions are proposed, and the performances of the solutions are compared. We also prove that one of the proposed algorithms gives a solution that, in the worst case, is at most 4/3 times the cost of the optimal solution.