Gonality of metric graphs and Catalan-many tropical morphisms to trees

This thesis consists of two points of view to regard degree-(g′+1) tropical morphisms Φ : (Γ,w) → Δ from a genus-(2g′) weighted metric graph (Γ,w) to a metric tree Δ, where g′ is a positive integer. The first point of view, developed in Part I, is purely combinatorial and constructive. It culminates with an application to bound the gonality of (Γ,w). The second point of view, developed in Part II, incorporates category theory to construct a unified framework under which both Φ and higher dimensional analogues can be understood. These higher dimensional analogues appear in the construction of a moduli space Gtrop/g→0,d parametrizing the tropical morphisms Φ, and a moduli spaceMtrop/g parametrizing the (Γ,w). There is a natural projection map Π : Gtrop/g→0,d →Mtrop/g that sends Φ : (Γ,w) → Δ to (Γ,w). The strikingly beautiful result is that when g = 2g′ and d = g′+1, the projection Π itself is an indexed branched cover, thus having the same nature as the maps Φ that are being parametrized. Moreover, fibres of Π have Catalan-many points. Each part has its own introduction that motivates and describes the problem from its own perspective. Part I and its introduction are based on two articles which are joint work with Jan Draisma. Part II contains material intended to be published as two articles. There is also a layman summary available at the beginning

A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis

Toward Controllable and Robust Surface Reconstruction from Spatial Curves

Reconstructing surface from a set of spatial curves is a fundamental problem in computer graphics and computational geometry. It often arises in many applications across various disciplines, such as industrial prototyping, artistic design and biomedical imaging. While the problem has been widely studied for years, challenges remain for handling different type of curve inputs while satisfying various constraints. We study studied three related computational tasks in this thesis. First, we propose an algorithm for reconstructing multi-labeled material interfaces from cross-sectional curves that allows for explicit topology control. Second, we addressed the consistency restoration, a critical but overlooked problem in applying algorithms of surface reconstruction to real-world cross-sections data. Lastly, we propose the Variational Implicit Point Set Surface which allows us to robustly handle noisy, sparse and non-uniform inputs, such as samples from spatial curves

Empirical evaluation of Soft Arc Consistency algorithms for solving Constraint Optimization Problems

A large number of problems in Artificial Intelligence and other areas of science can be viewed as special cases of constraint satisfaction or optimization problems. Various approaches have been widely studied, including search, propagation, and heuristics. There are still challenging real-world COPs that cannot be solved using current methods. We implemented and compared several consistency propagation algorithms, which include W-AC*2001, EDAC, VAC, and xAC. Consistency propagation is a classical method to reduce the search space in CSPs, and has been adapted to COPs. We compared several consistency propagation algorithms, based on the resemblance between the optimal value ordering and the approximate value ordering generated by them. The results showed that xAC generated value orderings of higher quality than W-AC*2001 and EDAC. We evaluated some novel hybrid methods for solving COPs. Hybrid methods combine consistency propagation and search in order to reach a good solution as soon as possible and prune the search space as much as possible. We showed that the hybrid method which combines the variant TP+OnOff and branch-and-bound search performed fewer constraint checks and searched fewer nodes than others in solving random and real-world COPs