Given a complete bipartite graph on two sets of points containing n points each, in a bottleneck matching problem, we want to find an one-to-one correspondence, also called a matching, that minimizes the length of its largest edge; the length of an edge is simply the Euclidean distance between its end-points. As an application, consider matching taxis to requests while minimizing the largest distance between any request to its matched taxi. The length of the largest edge (also called the bottleneck distance) has numerous applications in machine learning as well as topological data analysis. One can use the classical Hopcroft-Karp (HK-) Algorithm to find the bottleneck matching. In this thesis, we consider the case where A and B are points that are generated uniformly at random from a unit square. Instead of the classical HK-Algorithm, we implement and empirically analyze a new algorithm by Lahn and Raghvendra (Symposium on Computational Geometry, 2019). Our experiments show that our approach outperforms the HK-Algorithm based approach for computing bottleneck matching. / Master of Science / Consider the problem of matching taxis to an equal number of requests. While matching them, one objective is to minimize the largest distance between a request and its match. Finding such a matching is called the bottleneck matching problem. In addition, this optimization problem arises in topological data analysis as well as machine learning. In this thesis, I conduct an empirical analysis of a new algorithm, which is called the FAST-MATCH algorithm, to find the bottleneck matching. I find that, when a large input data is randomly generated from a unit square, the FAST-MATCH algorithm performs substantially faster than the classical methods.

While it has been known for some time that connected non-bipartite graphs have unique prime factorizations over the direct product, the same cannot be said of bipartite graphs. This is somewhat vexing, as bipartite graphs do have unique prime factorizations over other graph products (the Cartesian product, for example). However, it is fairly easy to show that a connected bipartite graph has only one prime bipartite factor, which begs the question: is such a prime bipartite factor unique? In other words, although a connected bipartite graph may have multiple prime factorizations over the direct product, do such factorizations contain the same prime bipartite factor? It has previously been shown by Hammack that when the prime bipartite factor is K_2, this is in fact true. The goal of this paper is to prove that this is in fact true for any prime bipartite factor, provided the graph being factored is R-thin. The proof of the main result takes the same initial approach as the proof by Hammack, before moving into new territory in order to prove the final result.

With rapid growth of information on the Web, Web search engines have gained great momentum for exploiting valuable Web resources. Although keywords-based Web search engines provide relevant search results in response to users’ queries, future enhancement is still needed. Three important issues include (1) search results can be diverse because ambiguous keywords in queries can be interpreted to different meanings; (2) indentifying keywords in long queries is difficult for search engines; and (3) generating query-specific Web page summaries is desirable for Web search results’ previews. Based on clickthrough data, this thesis proposes a query-concept bipartite graph for representing queries’ relations, and applies the queries’ relations to applications such as (1) personalized query suggestions, (2) long queries Web searches and (3) query-specific Web page summarization. Experimental results show that query-concept bipartite graphs are useful for performance improvement for the three applications.

A theorem of Edmonds characterizes when a pair of matroids has a common basis. Enumerating the common bases of a pair of matroid is a much harder problem, and includes the #P-complete problem of counting the number of perfect matchings in a bipartite graph. We focus on the problem of counting the common bases in pairs of regular matroids, and describe a class called <i>Pfaffian matroid pairs</i> for which this enumeration problem can be solved. We prove that when a pair of regular matroids is non-Pfaffian, there is a set of common bases which certifies this, and that the number of bases in the certificate is linear in the size of the ground set of the matroids. When both matroids in a pair are series-parallel, we prove that determining if the pair is Pfaffian is equivalent to finding an edge signing in an associated graph, and in the case that the pair is non-Pfaffian, we obtain a characterization of this associated graph. Pfaffian bipartite graphs are a class of graphs for which the number of perfect matchings can be determined; we show that the class of series-parallel Pfaffian matroid pairs is an extension of the class of Pfaffian bipartite graphs. Edmonds proved that the polytope generated by the common bases of a pair of matroids is equal to the intersection of the polytopes generated by the bases for each matroid in the pair. We consider when a similar property holds for the binary space, and give an excluded minor characterization of when the binary space generated by the common bases of two matroids can not be determined from the binary spaces for the individual matroids. As a result towards a description of the lattice of common bases for a pair of matroids, we show that the lattices for the individual matroids determine when all common bases of a pair of matroids intersect a subset of the ground set with fixed cardinality.

A theorem of Edmonds characterizes when a pair of matroids has a common basis. Enumerating the common bases of a pair of matroid is a much harder problem, and includes the #P-complete problem of counting the number of perfect matchings in a bipartite graph. We focus on the problem of counting the common bases in pairs of regular matroids, and describe a class called <i>Pfaffian matroid pairs</i> for which this enumeration problem can be solved. We prove that when a pair of regular matroids is non-Pfaffian, there is a set of common bases which certifies this, and that the number of bases in the certificate is linear in the size of the ground set of the matroids. When both matroids in a pair are series-parallel, we prove that determining if the pair is Pfaffian is equivalent to finding an edge signing in an associated graph, and in the case that the pair is non-Pfaffian, we obtain a characterization of this associated graph. Pfaffian bipartite graphs are a class of graphs for which the number of perfect matchings can be determined; we show that the class of series-parallel Pfaffian matroid pairs is an extension of the class of Pfaffian bipartite graphs. Edmonds proved that the polytope generated by the common bases of a pair of matroids is equal to the intersection of the polytopes generated by the bases for each matroid in the pair. We consider when a similar property holds for the binary space, and give an excluded minor characterization of when the binary space generated by the common bases of two matroids can not be determined from the binary spaces for the individual matroids. As a result towards a description of the lattice of common bases for a pair of matroids, we show that the lattices for the individual matroids determine when all common bases of a pair of matroids intersect a subset of the ground set with fixed cardinality.

Approximation algorithms are widely used for problems related to computational geometry, complex optimization problems, discrete min-max problems and NP-hard and space hard problems. Due to the complex nature of such problems, imperative languages are perhaps not the best-suited solution when it comes to their actual implementation. Functional languages like Haskell could be a good candidate for the aforementioned mentioned issues. Haskell is used in industries as well as in commercial applications, e.g., concurrent applications, statistics, symbolic math and financial analysis. Several approximation algorithms have been proposed for different problems that naturally arise in the DNA clone classifications. In this thesis, we have performed an initial and explorative study on applying functional languages for approximation algorithms. Specifically, we have implemented a well known approximate clustering algorithm both in Haskell and in Java and we discuss the suitability of applying functional languages for the implementation of approximation algorithms, in particular for graph theoretical approximate clustering problems with applications in DNA clone classification. We also further explore the characteristics of Haskell that makes it suitable for solving certain classes of problems that are hard to implement using imperative languages. / Muhammad Imran Shafi: 29A Sodergatan 19547 Marsta, 0737171514, Muhammad Akram C/O Saad Bin Azhar Folkparksvagen 20/10 Ronneby, 0762899111

A graph is an ordered pair composed by a set of vertices and a set of edges, the latter consisting of unordered pairs of vertices. Two vertices in such a pair are each others neighbors. Two edges are adjacent if they share a common vertex. Denote the amount of edges that share a specific vertex as the degree of the vertex. A proper edge-coloring of a graph is an assignment of colors from some finite set, to the edges of a graph where no two adjacent edges have the same color. A bipartition (X,Y) of a set of vertices V is an ordered pair of two disjoint sets of vertices such that V is the union of X and Y, where all the vertices in X only have neighbors in Y and vice versa. A bipartite graph is a graph whose vertices admit a bipartition (X,Y). Let G be one such graph. An X-interval coloring of G is a proper edge coloring where the colors of the edges incident to each vertex in X form an interval of integers. Denote by χ'int(G,X) the least number of colors needed for an X-interval coloring of G. In this paper we prove that if G is a bipartite graph with maximum degree 3n (n is a natural number), where all the vertices in X have degree 3, then <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathit%7B%5Cchi'_%7Bint%7D%5Cleft(G,X%5Cright)%5Cleq%7D%0A%5C%5C%0A%5Cmathit%7B%5Cleft(n-1%5Cright)%5Cleft(3n+5%5Cright)/2+3%7D%0A%5C%5C%0A%5Cmathit%7Bif%20n%20is%20odd,%7D%0A%5C%5C%0A%5Cmathit%7Bor%7D%0A%5C%5C%0A%5Cmathbf%7B3n%5E%7B2%7D/2+1%7D%0A%5C%5C%0A%5Cmathit%7Bif%20n%20is%20even%7D.%0A" />

Criminal networks exploit vulnerabilities in the global financial system, using it as a conduit to launder criminal proceeds. Law enforcement agencies, financial institutions, and regulatory organizations often scrutinize voluminous financial records for suspicious activities and criminal conduct as part of anti-money laundering investigations. However, such studies are narrowly focused on incidents and triggered by tip-offs rather than data mining insights. This research models cross-border financial flows using a network theoretic approach and proposes a symmetric-key encryption algorithm to preserve information privacy in multi-dimensional data sets. The newly developed tools will enable regulatory organizations, financial institutions, and law enforcement agencies to identify suspicious activity and criminal conduct in cross-border financial transactions. Anti-money laundering, which comprises laws, regulations, and procedures to combat money laundering, requires financial institutions to verify and identify their customers in various circumstances and monitor suspicious activity transactions. Instituting anti-money laundering laws and regulations in a country carries the benefit of creating a data-rich environment, thereby facilitating non-classical analytical strategies and tools. Graph theory offers an elegant way of representing cross-border payments/receipts between resident and non-resident parties (nodes), with links representing the parties' transactions. The network representations provide potent data mining tools, facilitating a better understanding of transactional patterns that may constitute suspicious transactions and criminal conduct. Using network science to analyze large and complex data sets to detect anomalies in the data set is fast becoming more important and exciting than merely learning about its structure. This research leverages advanced technology to construct and visualize the cross-border financial flows' network structure, using a directed and dual-weighted bipartite graph. Furthermore, the develops a centrality measure for the proposed cross-border financial flows network using a method based on matrix multiplication to answer the question, "Which resident/non-resident nodes are the most important in the cross-border financial flows network?" The answer to this question provides data mining insights about the network structure. The proposed network structure, centrality measure, and characterization using degree distributions can enable financial institutions and regulatory organizations to identify dominant nodes in complex multi-dimensional data sets. Most importantly, the results showed that the research provides transaction monitoring capabilities that allow the setting of customer segmentation criteria, complementing the built-in transaction-specific triggers methods for detecting suspicious activity transactions. / Thesis (PhD)--University of Pretoria, 2021. / Banking Sector Education and Training Authority (BANKSETA) / UP Postgraduate Bursary / Industrial and Systems Engineering / PhD / Unrestricted

Necessary and sufficient conditions are given for the existence of isomorphic decompositions of the complete bipartite graph, the complete graph with a hole, and the λ-fold complete graph into copies of a 4-cycle with a pendant edge.

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w).