148 research outputs found
Graph theoretic generalizations of clique: optimization and extensions
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
On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem
We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve the main open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number
Approximation methods in geometry and topology: learning, coarsening, and sampling
Data materialize in many different forms and formats. These can be continuous or
discrete, from algebraic expressions to unstructured pointclouds and highly structured graphs and simplicial complexes. Their sheer volume and plethora of different
modalities used to manipulate and understand them highlight the need for expressive abstractions and approximations, enabling novel insights and efficiency.
Geometry and topology provide powerful and intuitive frameworks for modelling
structure, form, and connectivity. Acting as a multi-focal lens, they enable inspection
and manipulation at different levels of detail, from global discriminant features to
local intricate characteristics. However, these fundamentally algebraic theories do
not scale well in the digital world.
Adjusting topology and geometry to the computational setting is a non-trivial task,
adhering to the âno free lunchâ adage. The necessary discretizations can be inaccurate, the underlying combinatorial structures can grow unmanageably in size, and
computing salient topological and geometric features can become computationally
taxing. Approximations are a necessity when theory cannot accommodate for efficient algorithms.
This thesis explores different approaches to simplifying computations pertaining to
geometry and topology via approximations. Our methods contribute to the approximation of topological features on discrete domains, and employ geometry and topology to efficiently guide discretizations and approximations. This line of work fits un der the umbrella of Topological Data Analysis (TDA) and Discrete Geometry, which
aim to bridge the continuous algebraic mindset with the discrete.
We construct topological and geometric approximation methods operating on three
different levels. We approximate topological features on discrete combinatorial spaces;
we approximate the combinatorial spaces themselves; and we guide processes that
allow us to discretize domains via sampling. With our Dist2Cycle model we learn geometric manifestations of topological features, the âoptimalâ homology generating
cycles. This is achieved by a novel simplicial complex neural network that exploits
the kernel of Hodge Laplacian operators to localize concise homology generators.
Compression of meshes and arbitrary simplicial complexes is made possible by our
general spectral coarsening strategy. Functional and structural properties are preserved by optimizing for important eigenspaces of general differential operators, the
Hodge Laplacians, at multiple dimensions. Finally, we offer a geometry-driven sampling strategy for data accumulation and stochastic integration. By employing the
kd-tree geometric partitioning algorithm we construct a sample set with provable
equidistribution guarantees.
Our findings are contextualized within prior and recent work, and our methods are
thoroughly discussed and evaluated on diverse settings. Ultimately, we are making
a claim towards the usefulness of examining the ever-present topological and geometric properties of data, not only in terms of feature discovery, but also as informed
generation, manipulation, and simplification tools
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
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Computational Geometric and Algebraic Topology
Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity.
At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field
Novel approaches for solving large-scale optimization problems on graphs
This dissertation considers a class of closely related NP-hard otpimization problems
on graphs that arise in many important applications, including network-based data
mining, analysis of the stock market, social networks, coding theory, fault diagnosis,
molecular biology, biochemistry and genomics. In particular, the problems of interest
include the classical maximum independent set problem (MISP) and maximum clique
problem (MCP), their vertex-weighted vesrions, as well as novel optimization models
that can be viewed as practical relaxations of their classical counterparts.
The concept of clique has been a popular instrument in analysis of networks, and
is, essentially, an idealized model of a âclosely connected groupâ, or a cluster. But,
at the same time, the restrictive nature of the definition of clique makes the clique
model impractical in many applications. This motivated the development of clique
relaxation models that relax different properties of a clique. On the one hand, while
still possessing some clique-like properties, clique relaxations are not as âperfectâ as
cliques; and on the other hand, they do not exhibit the disadvantages associated with
a clique. Using clique relaxations allows one to compromise between perfectness and
flexibility, between ideality and reality, which is a usual issue that an engineer deals
with when applying theoretical knowledge to solve practical problems in industry.
The clique relaxation models studied in this dissertation were first proposed in the
literature on social network analysis, however they have not been well investigated from a mathematical programming perspective.
This dissertation considers new techniques for solving the MWISP and clique
relaxation problems and investigates their effectiveness from theoretical and computational
perspectives. The main results obtained in this work include (i) developing a
scale-reduction approach for MWISP based on the concept of critical set and comparing
it theoretically with other approaches; (ii) obtaining theoretical complexity results
for clique relaxation problems; (iii) developing algorithms for solving the clique relaxation
problems exactly; (iv) carrying out computational experiments to demonstrate
the performance of the proposed approaches, and, finally, (v) applying the obtained
theoretical results to several real-life problems
In pursuit of linear complexity in discrete and computational geometry
Many computational problems arise naturally from geometric data. In this thesis, we consider three such problems: (i) distance optimization problems over point sets, (ii) computing contour trees over simplicial meshes, and (iii) bounding the expected complexity of weighted Voronoi diagrams. While these topics are broad, here the focus is on identifying structure which implies linear (or near linear) algorithmic and descriptive complexity.
The first topic we consider is in geometric optimization. More specifically, we define a large class of distance problems, for which we provide linear time exact or approximate solutions. Roughly speaking, the class of problems facilitate either clustering together close points (i.e. netting) or throwing out outliers (i.e pruning), allowing for successively smaller summaries of the relevant information in the input. A surprising number of classical geometric optimization problems are unified under this framework, including finding the optimal k-center clustering, the kth ranked distance, the kth heaviest edge of the MST, the minimum radius ball enclosing k points, and many others. In several cases we get the first known linear time approximation algorithm for a given problem, where our approximation ratio matches that of previous work.
The second topic we investigate is contour trees, a fundamental structure in computational topology. Contour trees give a compact summary of the evolution of level sets on a mesh, and are typically used on massive data sets. Previous algorithms for computing contour trees took Î(n log n) time and were worst-case optimal. Here we provide an algorithm whose running time lies between Î(nα(n)) and Î(n log n), and varies depending on the shape of the tree, where α(n) is the inverse Ackermann function. In particular, this is the first algorithm with O(nα(n)) running time on instances with balanced contour trees. Our algorithmic results are complemented by lower bounds indicating that, up to a factor of α(n), on all instance types our algorithm performs optimally.
For the final topic, we consider the descriptive complexity of weighted Voronoi diagrams. Such diagrams have quadratic (or higher) worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis. A new diagram, called the candidate diagram, is introduced, which allows us to bound the complexity of weighted Voronoi diagrams arising from a particular probabilistic input model. Specifically, we assume weights are randomly permuted among fixed Voronoi sites, an assumption which is weaker than the more typical sampled locations assumption. Under this assumption, the expected complexity is shown to be near linear
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