Abstract: We consider the Railway Traveling Salesman Problem. We
show that this problem can be reduced to a variant of the generalized
traveling salesman problem, defined on an undirected graph G = (V,E)
with the nodes partitioned into clusters, which consists in finding a mini-
mum cost cycle spanning a subset of nodes with the property that exactly
two nodes are chosen from each cluster. We describe an exact exponen-
tial time algorithm for the problem, as well we present two mixed integer
programming models of the problem. Based on one of this models pro-
posed, we present an efficient solution procedure based on a cutting plane
algorithm. Extensive computational results for instances taken from the
railroad company of the Netherlands Nederlandse Spoorwegen and involv-
ing graphs with up to 2182 nodes and 38650 edges are reported.

Abstract: Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar, and sparse networks. The approach used is to preprocess the inputn-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after anO(n log n) preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in timeO(n2). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, \~{a}, of the input network. The parameter \~{a} varies between 1 and {\`E}(n); the algorithms perform well when \~{a} = o(n). The value of a min-cut can be found in timeO(n + \~{a}2log \~{a}) and all-pairs min-cut can be solved in timeO(n2 + \~{a}4log \~{a}) for sparse networks. The corresponding running times for planar networks areO(n + \~{a} log \~{a}) andO(n2 + \~{a}3log \~{a}), respectively. The latter bounds depend on a result of independent interest; outerplanar networks have small “mimicking” networks that are also outerplanar.

Abstract: In this Phd thesis,, we try to use formal logic and threshold phenomena that asymptotically emerge with certainty in order to build new trust models and to evaluate the existing one. The departure point of our work is that dynamic, global computing systems are not amenable to a static viewpoint of the trust concept, no matter how this concept is formalized. We believe that trust should be a statistical, asymptotic concept to be studied in the limit as the system's components grow according to some growth rate. Thus, our main goal is to define trust as an emerging system property that ``appears'' or "disappears" when a set of properties hold, asymptotically with probability$ 0$ or $1$ correspondingly . Here we try to combine first and second order logic in order to analyze the trust measures of specific network models. Moreover we can use formal logic in order to determine whether generic reliability trust models provide a method for deriving trust between peers/entities as the network's components grow. Our approach can be used in a wide range of applications, such as monitoring the behavior of peers, providing a measure of trust between them, assessing the level of reliability of peers in a network. Wireless sensor networks are comprised of a vast number of ultra-small autonomous computing, communication and sensing devices, with restricted energy and computing capabilities, that co-operate to accomplish a large sensing task. Sensor networks can be very useful in practice. Such systems should at least guarantee the confidentiality and integrity of the information reported to the controlling authorities regarding the realization of environmental events. Therefore, key establishment is critical for the protection in wireless sensor networks and the prevention of adversaries from attacking the network. Finally in this dissertation we also propose three distributed group key establishment protocols suitable for such energy constrained networks. This dissertation is composed of two parts. Part I develops the theory of the first and second order logic of graphs - their definition, and the analysis of their properties that are expressible in the {\em first order language} of graphs. In part II we introduce some new distributed group key establishment protocols suitable for sensor networks. Several key establishment schemes are derived and their performance is demonstrated.

Abstract: Random Intersection Graphs, Gn,m,p, is a class of random graphs introduced in Karoński (1999) [7] where each of the n vertices chooses independently a random subset of a universal set of m elements. Each element of the universal sets is chosen independently by some vertex with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=left ceilingn{\'a}right ceiling, for any real {\'a} different than one, we establish here, for the first time, a sharp threshold for the graphproperty “Contains a Hamilton cycle”. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection graph model.

Abstract: We work on an extension of the Population Protocol model of Angluin et al. that allows edges of the communication graph, G, to have states that belong to a constant size set. In this extension, the so called Mediated Population Protocol model (MPP), both uniformity and anonymity are preserved. We study here a simplified version of MPP in order to capture MPP's ability to stably compute graph properties. To understand properties of the communication graph is an important step in almost any distributed system. We prove that any graphproperty is not computable if we allow disconnected communication graphs. As a result, we focus on studying (at least) weakly connected communication graphs only and give several examples of computable properties in this case. To do so, we also prove that the class of computable properties is closed under complement, union and intersection operations. Node and edge parity, bounded out-degree by a constant, existence of a node with more incoming than outgoing neighbors, and existence of some directed path of length at least k=O(1) are some examples of properties whose computability is proven. Finally, we prove the existence of symmetry in two specific communication graphs and, by exploiting this, we prove that there exists no protocol, whose states eventually stabilize, to determine whether G contains some directed cycle of length 2.

Abstract: Motivated by the problem of efficient sensor network data collection via a mobile sink, we present undergoing research in accelerated random walks on Random Geometric Graphs. We first propose a new type of random walk, called the {\'a}-stretched random walk, and compare it to three known random walks. We also define a novel performance metric called Proximity Cover Time which, along with other metrics such us visit overlap statistics and proximity variation, we use to evaluate the performance properties and features of the various walks. Finally, we present future plans on investigating a relevant combinatorial property of Random Geometric Graphs that may lead to new, faster random walks and metrics.