Detecting critical nodes in sparse graphs
Computers and Operations Research
A Branch and Cut solver for the maximum stable set problem
Journal of Combinatorial Optimization
Weighted graph-based methods for identifying the most influential actors in trust social networks
International Journal of Networking and Virtual Organisations
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This dissertation develops novel approaches to solving computationally difficult combinatorial optimization problems on graphs, namely maximum independent set, maximum clique, graph coloring, minimum dominating sets and related problems. The application areas of the considered problems include information retrieval, classification theory, economics, scheduling, experimental design, and computer vision among many others. The maximum independent set and related problems are formulated as nonlinear programs, and new methods for finding good quality approximate solutions in reasonable computational times are introduced. All algorithms are implemented and successfully tested on a number of examples from diverse application areas. The proposed methods favorably compare with other competing approaches. A large part of this dissertation is devoted to detailed studies of selected applications of the problems of interest. Novel techniques for analyzing the structure of financial markets based on their network representation are proposed and verified using massive data sets generated by the U.S. stock markets. The network representation of a market is based on cross-correlations of price fluctuations of the financial instruments, and provides a valuable tool for classifying the instruments. In another application, new exact values and estimates of size of the largest error correcting codes are computed using optimization in specially constructed graphs. Error correcting codes lie in the heart of digital technology, making cell phones, compact disk players and modems possible. They are also of a special significance due to increasing importance of reliability issues in Internet transmissions. Finally, efficient approximate algorithms for construction of virtual backbone in wireless networks by means of solving the minimum connected dominating set problem in unit-disk graphs are developed and tested.