Given a graph, a matching is a set of vertex-disjoint edges. Graph matchings have been well studied, since they play a fundamental role in algorithmic theory as well as motivate many practical applications. Of particular interest is the problem of finding a maximum cardinality matching of a graph. Also of interest is the weighted variant: the problem of computing a minimum-cost maximum cardinality matching. For an arbitrary graph with m edges and n vertices, there are known, long-standing combinatorial algorithms that compute a maximum cardinality matching in O(m\sqrt{n}) time. For graphs with non-negative integer edge costs at most C, it is known how to compute a minimum-cost maximum cardinality matching in roughly O(m\sqrt{n} log(nC)) time using combinatorial methods. While non-combinatorial methods exist, they are generally impractical and not well understood due to their complexity. As a result, there is great interest in obtaining faster matching algorithms that are purely combinatorial in nature. Improving existing combinatorial algorithms for arbitrary graphs is considered to be a very difficult problem. To make the problem more approachable, it is desirable to make some additional assumptions about the graph. For our work, we make two such assumptions. First, we assume the graph is bipartite. Second, we assume that the graph has a small balanced separator, meaning it is possible to split the graph into two roughly equal-size components by removing a relatively small portion of the graph. Several well-studied classes of graphs have separator-like properties, including planar graphs, minor-free graphs, and geometric graphs. For such graphs, we describe a framework, a general set of techniques for designing efficient algorithms. We demonstrate this framework by applying it to yield polynomial-factor improvements for several open-problems in bipartite matching. / Doctor of Philosophy / Assume we are given a list of objects, and a list of compatible pairs of these objects. A matching consists of a chosen subset of these compatible pairs, where each object participates in at most one chosen pair. For any chosen pair of objects, we say the these two objects are matched. Generally, we seek to maximize the number of compatible matches. A maximum cardinality matching is a matching with the largest possible size. In many cases, there are multiple options for maximizing the number of compatible pairings. While maximizing the size of the matching is often the primary concern, one may also seek to minimize the cost of the matching. This is known as the minimum-cost maximum-cardinality matching problem. These two matching problems have been well studied, since they play a fundamental role in algorithmic theory as well as motivate many practical applications. Our interest is in the design of algorithms for both of these problems that are efficiently scalable, even as the number of objects involved grows very large. To aid in the design of scalable algorithms, we observe that some inputs have good separators, meaning that by removing some subset S of objects, one can divide the remaining objects into two sets V and V', where all pairs of objects between V and V' are incompatible. We design several new algorithms that exploit good separators, and prove that these algorithms scale better than previously existing approaches.

Biological network data is plentiful with practically every experimental methodology giving 'network views' into cellular function and behavior. Bioinformatic screens that yield network data include, for example, genome-wide deletion screens, protein-protein interaction assays, RNA interference experiments, and methods to probe metabolic pathways. Efficient and comprehensive computational approaches are required to model these screens and gain insight into the nature of biological networks. This thesis presents three new algorithms to model and mine network datasets. First, we present an algorithm that models genome-wide perturbation screens by deriving relations between phenotypes and subsequently using these relations in a local manner to derive genephenotype relationships. We show how this algorithm outperforms all previously described algorithms for gene-phenotype modeling. We also present theoretical insight into the convergence and accuracy properties of this approach. Second, we define a new data mining problem–constrained minimal separator mining—and propose algorithms as well as applications to modeling gene perturbation screens by viewing the perturbed genes as a graph separator. Both of these data mining applications are evaluated on network datasets from S. cerevisiae and C. elegans. Finally, we present an approach to model the relationship between metabolic pathways and operon structure in prokaryotic genomes. In this approach, we present a new pattern class—biclusters over domains with supplied partial orders—and present algorithms for systematically detecting such biclusters. Together, our data mining algorithms provide a comprehensive arsenal of techniques for modeling gene perturbation screens and metabolic pathways. / Ph. D.