We introduce a very simple but efficient idea for branch & bound (B&B) algorithms in global optimization (GO). As input for our generic algorithm, we need an upper bound algorithm for the GO maximization problem and a branching rule. The latter reduces the problem into several smaller subproblems of the same type. The new B&B approach delivers one global optimizer or, if stopped before finished, improved upper and lower bounds for the problem. Its main difference to commonly used B&B techniques is its ability to approximate the problem from above and from below while traversing the problem tree. It needs no supplementary information about the system optimized and does not consume more time than classical B&B techniques. Experimental results with the maximum clique problem illustrate the benefit of this new method. (author's abstract) / Series: Working Papers on Information Systems, Information Business and Operations

Graph theory is fascinating branch of math. Leonhard Euler introduced the concept of Graph Theory in his paper about the seven bridges of Konigsberg published in 1736. In a nutshell, graph theory is the study of pair-wise relationships between objects. Each object is represented using a vertex and in case of a relationship between a pair of vertices, they will be connected using an edge. In this dissertation, graph theory is used to study several important combinatorial optimization problems. In chapter 2, we study the multi-dimensional assignment problem using their underlying hypergraphs. It will be shown how the MAP can be represented by a k-partite graph and how any solution to MAP is associated to a k-clique in the respective k-partite graph. Two heuristics are proposed to solve the MAP and computational studies are performed to compare the performance of the proposed methods with exact solutions. On the heels of chapter 3, a new branch-and-bound method is proposed to solve the problem of finding all k-cliques in a k-partite graph. The new method utilizes bitsets as the data-structure to represent graph data. A new pruning method is introduced in BitCLQ, and CPU instructions are used to improve the performance of the branch-and-bound method. BitCLQ gains up to 300% speed up over existing methods. In chapter 4, two new heuristic to solve decomposable cost MAP's are proposed. The proposed heuristic are based on the partitioning of the underlying graph representing the MAP. In the first heuristic method, MAP's are partitioned into several smaller MAP's with the same dimensiality and smaller cardinality. The solution to the original MAP is constructed incrementally, by using the solutions obtained from each of the smaller MAP's. The second heuristic works in the same fashion. But instead of partitioning the graph along the cardinality, graphs are divided into smaller graphs with the same cardinality but smaller dimensionality. The result of each heuristic is then compared with a well-known existing heuristic. An important problem in graph theory is the maximum clique problem (MCQ). A clique in a graph is defined as a complete subgraph. MCQ problem entails finding the size of the largest clique contained in a graph. General branch-and-bound methods to solve MCQ use graph coloring to find an upper bound on the size of the maximum clique. Recently, a new MAX-SAT based branch-and-bound method for MCQ is proposed that improves the quality of the upper bound obtained from graph coloring. In chapter 5, a branch and bound algorithm is proposed for the maximum clique problem. The proposed method uses bitsets as the data-structure. The result of the computational studies to compare the proposed method with existing methods for MCQ is provided. Chapter 6 contains an application of a graph theory in solving a risk management problem. A new mixed-integer mathematical model to formulate a risk-based network is provided. It will be shown that an optimal solution of the model is a maximal clique in the underlying graph representing the network. The model is then solved using a graph-theoretic approach and the results are compared to CPLEX.

We introduce and study a novel graph optimization problem to search for multiple cliques with the maximum overall weight, to which we denote as the Maximum Weighted Multiple Clique Problem (MWMCP). This problem arises in research involving network-based data mining, specifically, in bioinformatics where complex diseases, such as various types of cancer and diabetes, are conjectured to be triggered and influenced by a combination of genetic and environmental factors. To integrate potential effects from interplays among underlying candidate factors, we propose a new network-based framework to identify effective biomarkers by searching for "groups" of synergistic risk factors with high predictive power to disease outcome. An interaction network is constructed with vertex weight representing individual predictive power of candidate factors and edge weight representing pairwise synergistic interaction among factors. This network-based biomarker identification problem is then formulated as a MWMCP. To achieve near optimal solutions for large-scale networks, an analytical algorithm based on column generation method as well as a fast greedy heuristic have been derived. Also, to obtain its exact solutions, an advanced branch-price-and-cut algorithm is designed and solved after studying the properties of the problem. Our algorithms for MWMCP have been implemented and tested on random graphs and promising results have been obtained. They also are used to analyze two biomedical datasets: a Type 1 Diabetes (T1D) dataset from the Diabetes Prevention Trial-Type 1 (DPT-1) Study, and a breast cancer genomics dataset for metastasis prognosis. The results demonstrate that our network-based methods can identify important biomarkers with better prediction accuracy compared to the conventional feature selection that only considers individual effects.

This dissertation considers graph theoretic generalizations of the maximum clique problem. Models that were originally proposed in social network analysis literature, are investigated from a mathematical programming perspective for the first time. A social network is usually represented by a graph, and cliques were the first models of "tightly knit groups" in social networks, referred to as cohesive subgroups. Cliques are idealized models and their overly restrictive nature motivated the development of clique relaxations that relax different aspects of a clique. Identifying large cohesive subgroups in social networks has traditionally been used in criminal network analysis to study organized crimes such as terrorism, narcotics and money laundering. More recent applications are in clustering and data mining wireless networks, biological networks as well as graph models of databases and the internet. This research has the potential to impact homeland security, bioinformatics, internet research and telecommunication industry among others. The focus of this dissertation is a degree-based relaxation called k-plex. A distance-based relaxation called k-clique and a diameter-based relaxation called k-club are also investigated in this dissertation. We present the first systematic study of the complexity aspects of these problems and application of mathematical programming techniques in solving them. Graph theoretic properties of the models are identified and used in the development of theory and algorithms. Optimization problems associated with the three models are formulated as binary integer programs and the properties of the associated polytopes are investigated. Facets and valid inequalities are identified based on combinatorial arguments. A branch-and-cut framework is designed and implemented to solve the optimization problems exactly. Specialized preprocessing techniques are developed that, in conjunction with the branch-and-cut algorithm, optimally solve the problems on real-life power law graphs, which is a general class of graphs that include social and biological networks. Computational experiments are performed to study the effectiveness of the proposed solution procedures on benchmark instances and real-life instances. The relationship of these models to the classical maximum clique problem is studied, leading to several interesting observations including a new compact integer programming formulation. We also prove new continuous non-linear formulations for the classical maximum independent set problem which maximize continuous functions over the unit hypercube, and characterize its local and global maxima. Finally, clustering and network design extensions of the clique relaxation models are explored.