Towards Distributed Two-Stage Stochastic Optimization

The weighted vertex cover problem is concerned with selecting a subset of the vertices that covers a target set of edges with the objective of minimizing the total cost of the selected vertices. We consider a variant of this classic combinatorial optimization problem where the target edge set is not fully known; rather, it is characterized by a probability distribution. Adhering to the model of two-stage stochastic optimization, the execution is divided into two stages so that in the first stage, the decision maker selects some of the vertices based on the probabilistic forecast of the target edge set. Then, in the second stage, the edges in the target set are revealed and in order to cover them, the decision maker can augment the vertex subset selected in the first stage with additional vertices. However, in the second stage, the vertex cost increases by some inflation factor, so the second stage selection becomes more expensive. The current paper studies the two-stage stochastic vertex cover problem in the realm of distributed graph algorithms, where the decision making process (in both stages) is distributed among the vertices of the graph. By combining the stochastic optimization toolbox with recent advances in distributed algorithms for weighted vertex cover, we develop an algorithm that runs in time O(log (?) / ?), sends O(m) messages in total, and guarantees to approximate the optimal solution within a (3 + ?)-ratio, where m is the number of edges in the graph, ? is its maximum degree, and 0 < ? < 1 is a performance parameter

Extremal and probabilistic results for regular graphs

In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness. We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory. We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs. Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles

Magneto-inductive wireless underground sensor networks: novel longevity model, communication concepts and workarounds to key theoretical issues using analogical thinking

This research has attempted to devise novel workarounds to key theoretical issues in magneto-inductive wireless underground sensor networks (WUSNs), founded on analogical thinking (Gassmann & Zeschky 2008). The problem statement for this research can be summarized as follows. There has been a substantial output of research publications in the past 5 years, devoted to theoretically analysing and resolving the issues pertaining to deployment of MI based WUSNs. However, no alternate solution approaches to such theoretical analyses have been considered. The goal of this research was to explore such alternate solution approaches. This research has used the principle of analogical thinking in devising such alternate solution approaches. This research has made several key contributions to the existing body of work. First, this research is the first of its kind to demonstrate by means of review of state-of-the-art research on MI based WUSNs, the largely theoretical genus of the research to the exclusion of alternate solution approaches to circumvent key theoretical issues. Second, this research is the first of its kind to introduce the notion of analogical thinking as a solution approach in finding viable workarounds to theoretical impediments in MI based WUSNs, and validate such solution approach by means of simulations. Third, this research is the first of its kind to explore novel communication concepts in the realm of MI based WUSNs, based on analogical thinking. Fourth, this research is the first of its kind to explore a novel longevity model in the realm of MI based WUSNs, based on analogical thinking. Fifth, this research is also the first to extend the notion of analogical thinking to futuristic directions in MI based WUSNs research, by means of providing possible indicators drawn from various other areas of contemporary research. In essence, the author believes that the findings of this research mark a paradigm shift in the research on MI based WUSNs

Utilising restricted for-loops in genetic programming

Genetic programming is an approach that utilises the power of evolution to allow computers to evolve programs. While loops are natural components of most programming languages and appear in every reasonably-sized application, they are rarely used in genetic programming. The work is to investigate a number of restricted looping constructs to determine whether any significant benefits can be obtained in genetic programming. Possible benefits include: Solving problems which cannot be solved without loops, evolving smaller sized solutions which can be more easily understood by human programmers and solving existing problems quicker by using fewer evaluations. In this thesis, a number of explicit restricted loop formats were formulated and tested on the Santa Fe ant problem, a modified ant problem, a sorting problem, a visit-every-square problem and a difficult object classificat ion problem. The experimental results showed that these explicit loops can be successfully used in genetic programming. The evolutionary process can decide when, where and how to use them. Runs with these loops tended to generate smaller sized solutions in fewer evaluations. Solutions with loops were found to some problems that could not be solved without loops. The results and analysis of this thesis have established that there are significant benefits in using loops in genetic programming. Restricted loops can avoid the difficulties of evolving consistent programs and the infinite iterations problem. Researchers and other users of genetic programming should not be afraid of loops

Describing function theory as applied to thermal and neutronic problems

Describing functions have traditionally been used to obtain the solutions of systems of ordinary differential equations. In this work the describing function concept has been extended to include nonlinear, distributed parameter partial differential equations. A three-stage solution algorithm is presented which can be applied to any nonlinear partial differential equation. Two generalized integral transforms were developed as the T-transform for the time domain and the B-transform for the spatial domain. As specific applications of the solution technique two cases are considered: the heat conduction equation and the neutron diffusion equation. The thermal diffusion describing function (TDDF) is developed for conduction of heat in solids and a general iterative solution along with convergence criteria is presented. The proposed solution method is used to solve the problem of heat transfer in nuclear fuel rods with annular fuel pellets. As a special instance the solid cylindrical fuel pellet is examined. A computer program is written which uses the describing function concept for computing fuel pin temperatures in the radial direction during reactor transients. It was found that the quasi-linear method used in the describing function method is as accurate as nonlinear treatments and as fast as true linearization methods. The second problem investigated was the neutron diffusion equation which is intrinsically different from the first case. Although, for most situations, it can be treated as a linear differential equation, the describing function method is still applicable. A describing function solution is derived for two possible cases: constant diffusion coefficient and variable diffusion coefficient. Two classes of describing functions are defined for each case which portray the leakage and absorption phenomena. A study of the convergence criteria is also included. For the specific case of a slab reactor criticality problem the comparison between analytical and describing function solutions revealed an excellent agreement