823 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Engineering a Preprocessor for Symmetry Detection
State-of-the-art solvers for symmetry detection in combinatorial objects are becoming increasingly sophisticated software libraries. Most of the solvers were initially designed with inputs from combinatorics in mind (nauty, bliss, Traces, dejavu). They excel at dealing with a complicated core of the input. Others focus on practical instances that exhibit sparsity. They excel at dealing with comparatively easy but extremely large substructures of the input (saucy). In practice, these differences manifest in significantly diverging performances on different types of graph classes.
We engineer a preprocessor for symmetry detection. The result is a tool designed to shrink sparse, large substructures of the input graph. On most of the practical instances, the preprocessor improves the overall running time significantly for many of the state-of-the-art solvers. At the same time, our benchmarks show that the additional overhead is negligible.
Overall we obtain single algorithms with competitive performance across all benchmark graphs. As such, the preprocessor bridges the disparity between solvers that focus on combinatorial graphs and large practical graphs. In fact, on most of the practical instances the combined setup significantly outperforms previous state-of-the-art
Improving the Academic Success of Technical College Students with Disabilities: A Multisite Descriptive Case Study
Students with disabilities in higher education have lower retention and graduation rates than students without disabilities. While postsecondary administrators are attempting to meet the needs of students by implementing necessary reforms, barriers remain like issues with disclosure, transition planning, and faculty knowledge. This present qualitative descriptive case study sought to explore the instructional practices that were implemented by technical college educators to accommodate students with learning challenges, including students with disabilities, utilizing the Universal Design for Learning framework to determine which current technical college faculty instructional accommodations practices intersect with or diverge from Universal Design for Learning principles. The participants were a purposeful sample of 12 full-time technical college faculty members from six technical colleges in a southern state with at least five years of teaching experience at the postsecondary level and had worked with at least one student with a disability. Data were collected in three phases through the Universal Design for Learning Checklist, Semi-structured Interviews, and Document Analysis of course syllabi. Frequency counts and thematic analysis were utilized to analyze the data. This qualitative research has implications for identifying consistent and best instructional practices that positively impact the academic achievement of college students with disabilities. The findings indicated that technical college faculty have been implementing Universal Design for Learning instructional strategies, both intentionally and unknowingly, in an attempt to provide equitable access to all students regardless of ability and that technical college students can benefit from the implementation of Universal Design for Learning principles into college courses. The findings also implied that professional development training can become a vital aspect of instructors\u27 improvement programs to enlighten them about strategies that are available to improve their work with students with disabilities
Rainbow bases in matroids
Recently, it was proved by B\'erczi and Schwarcz that the problem of
factorizing a matroid into rainbow bases with respect to a given partition of
its ground set is algorithmically intractable. On the other hand, many special
cases were left open.
We first show that the problem remains hard if the matroid is graphic,
answering a question of B\'erczi and Schwarcz. As another special case, we
consider the problem of deciding whether a given digraph can be factorized into
subgraphs which are spanning trees in the underlying sense and respect upper
bounds on the indegree of every vertex. We prove that this problem is also
hard. This answers a question of Frank.
In the second part of the article, we deal with the relaxed problem of
covering the ground set of a matroid by rainbow bases. Among other results, we
show that there is a linear function such that every matroid that can be
factorized into bases for some can be covered by rainbow
bases if every partition class contains at most 2 elements
Algorithms Transcending the SAT-Symmetry Interface
Dedicated treatment of symmetries in satisfiability problems (SAT) is
indispensable for solving various classes of instances arising in practice.
However, the exploitation of symmetries usually takes a black box approach.
Typically, off-the-shelf external, general-purpose symmetry detection tools are
invoked to compute symmetry groups of a formula. The groups thus generated are
a set of permutations passed to a separate tool to perform further analyzes to
understand the structure of the groups. The result of this second computation
is in turn used for tasks such as static symmetry breaking or dynamic pruning
of the search space. Within this pipeline of tools, the detection and analysis
of symmetries typically incurs the majority of the time overhead for symmetry
exploitation.
In this paper we advocate for a more holistic view of what we call the
SAT-symmetry interface. We formulate a computational setting, centered around a
new concept of joint graph/group pairs, to analyze and improve the detection
and analysis of symmetries. Using our methods, no information is lost
performing computational tasks lying on the SAT-symmetry interface. Having
access to the entire input allows for simpler, yet efficient algorithms.
Specifically, we devise algorithms and heuristics for computing finest direct
disjoint decompositions, finding equivalent orbits, and finding natural
symmetric group actions. Our algorithms run in what we call
instance-quasi-linear time, i.e., almost linear time in terms of the input size
of the original formula and the description length of the symmetry group
returned by symmetry detection tools. Our algorithms improve over both
heuristics used in state-of-the-art symmetry exploitation tools, as well as
theoretical general-purpose algorithms
On Pairwise Graph Connectivity
A graph on at least k+1 vertices is said to have global connectivity k if any two of its vertices are connected by k independent paths. The local connectivity of two vertices is the number of independent paths between those specific vertices. This dissertation is concerned with pairwise connectivity notions, meaning that the focus is on local connectivity relations that are required for a number of or all pairs of vertices. We give a detailed overview about how uniformly k-connected and uniformly k-edge-connected graphs are related and provide a complete constructive characterization of uniformly 3-connected graphs, complementing classical characterizations by Tutte. Besides a tight bound on the number of vertices of degree three in uniformly 3-connected graphs, we give results on how the crossing number and treewidth behaves under the constructions at hand. The second central concern is to introduce and study cut sequences of graphs. Such a sequence is the multiset of edge weights of a corresponding Gomory-Hu tree. The main result in that context is a constructive scheme that allows to generate graphs with prescribed cut sequence if that sequence satisfies a shifted variant of the classical Erdős-Gallai inequalities. A complete characterization of realizable cut sequences remains open. The third central goal is to investigate the spectral properties of matrices whose entries represent a graph's local connectivities. We explore how the spectral parameters of these matrices are related to the structure of the corresponding graphs, prove bounds on eigenvalues and related energies, which are sums of absolute values of all eigenvalues, and determine the attaining graphs. Furthermore, we show how these results translate to ultrametric distance matrices and touch on a Laplace analogue for connectivity matrices and a related isoperimetric inequality
Mathematical Methods and Operation Research in Logistics, Project Planning, and Scheduling
In the last decade, the Industrial Revolution 4.0 brought flexible supply chains and flexible design projects to the forefront. Nevertheless, the recent pandemic, the accompanying economic problems, and the resulting supply problems have further increased the role of logistics and supply chains. Therefore, planning and scheduling procedures that can respond flexibly to changed circumstances have become more valuable both in logistics and projects. There are already several competing criteria of project and logistic process planning and scheduling that need to be reconciled. At the same time, the COVID-19 pandemic has shown that even more emphasis needs to be placed on taking potential risks into account. Flexibility and resilience are emphasized in all decision-making processes, including the scheduling of logistic processes, activities, and projects
Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization
Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes
General Course Catalog [2022/23 academic year]
General Course Catalog, 2022/23 academic yearhttps://repository.stcloudstate.edu/undergencat/1134/thumbnail.jp
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