Solving Hard Graph Problems with Combinatorial Computing and Optimization

Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are $NP$-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number $F_e(3,3;4)$, defined as the smallest order of a $K_4$-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were $19 leq F_e(3,3;4) leq 941$. We improve the upper bound to $F_e(3,3;4) leq 786$ using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number $R(C_4,K_m)$, which is the smallest $n$ such that any $n$-vertex graph contains a cycle of length four or an independent set of order $m$. With the help of combinatorial algorithms, we determine $R(C_4,K_9)=30$ and $R(C_4,K_{10})=36$ using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic

Graph dominance by rook domains for Znp and Zn3 × Zm2 graphs

Described within is the problem of finding near-minimum dominating subsets of agiven graph by rook domains. Specifically, we study the graphs of the kind ZnpandZn3×Zm2and introduce a simulated annealing algorithm to compute upper bounds ofthe size of minimum dominating subsets.We demonstrate the effectiveness of the algorithm by comparing the results with apreviously studied class of graphs, including the so-called “football pool” graphs andothers. We give some new upper bounds for graphs of the kind Znp, with p 4. Thecodes of some dominating subsets are given in an appendix.Keywords: Graph domination, simulated annealing, football pool problem, combinatorics.En este art´?culo se describe el problema de la dominaci´on de los grafos del tipo Znpy mezclas del tipo Zn3×Zm2a trav´es de subconjuntos dominantes de v´ertices de tama˜nom´?nimo. Se introduce un algoritmo del tipo de recocido simulado para calcular cotassuperiores de la cardinalidad de estos subconjuntos dominantes minimales.Se demuestra la eficiencia del algoritmo al comparar los resultados obtenidos conlos ya conocidos correspondientes a algunas clases de grafos, entre ellos los llamadosgrafos del “football pool problem”. Se establecen cotas superiores en algunos de losgrafos del tipo Znp, con p 4. Los c´odigos de algunos subconjuntos dominantes seincluyen en un ap´endice.Palabras clave: Dominaci´on de grafos, recocido simulado, problema de las apuestas enf´utbol, combinatoria

Real-time algorithm configuration

This dissertation presents a number of contributions to the field of algorithm configur- ation. In particular, we present an extension to the algorithm configuration problem, real-time algorithm configuration, where configuration occurs online on a stream of instances, without the need for prior training, and problem solutions are returned in the shortest time possible. We propose a framework for solving the real-time algorithm configuration problem, ReACT. With ReACT we demonstrate that by using the parallel computing architectures, commonplace in many systems today, and a robust aggregate ranking system, configuration can occur without any impact on performance from the perspective of the user. This is achieved by means of a racing procedure. We show two concrete instantiations of the framework, and show them to be on a par with or even exceed the state-of-the-art in offline algorithm configuration using empirical evaluations on a range of combinatorial problems from the literature. We discuss, assess, and provide justification for each of the components used in our framework instantiations. Specifically, we show that the TrueSkill ranking system commonly used to rank players’ skill in multiplayer games can be used to accurately es- timate the quality of an algorithm’s configuration using only censored results from races between algorithm configurations. We confirm that the order that problem instances arrive in influences the configuration performance and that the optimal selection of configurations to participate in races is dependent on the distribution of the incoming in- stance stream. We outline how to maintain a pool of quality configurations by removing underperforming configurations, and techniques to generate replacement configurations with minimal computational overhead. Finally, we show that the configuration space can be reduced using feature selection techniques from the machine learning literature, and that doing so can provide a boost in configuration performance

The BG News October 6, 1987

The BGSU campus student newspaper October 6, 1987. Volume 70 - Issue 25https://scholarworks.bgsu.edu/bg-news/5700/thumbnail.jp

Central Florida Future, Vol. 36 No. 36, January 22, 2004

Second rally takes aim at administrators\u27 actions; Board confronts gay discrimination; Violations could revolutionize combat; Faculty discuss forgiving bad grades; housing rates will jump but not too high.https://stars.library.ucf.edu/centralfloridafuture/2728/thumbnail.jp

Applied Metaheuristic Computing

For decades, Applied Metaheuristic Computing (AMC) has been a prevailing optimization technique for tackling perplexing engineering and business problems, such as scheduling, routing, ordering, bin packing, assignment, facility layout planning, among others. This is partly because the classic exact methods are constrained with prior assumptions, and partly due to the heuristics being problem-dependent and lacking generalization. AMC, on the contrary, guides the course of low-level heuristics to search beyond the local optimality, which impairs the capability of traditional computation methods. This topic series has collected quality papers proposing cutting-edge methodology and innovative applications which drive the advances of AMC

An investigation of multi-objective hyper-heuristics for multi-objective optimisation

In this thesis, we investigate and develop a number of online learning selection choice function based hyper-heuristic methodologies that attempt to solve multi-objective unconstrained optimisation problems. For the first time, we introduce an online learning selection choice function based hyperheuristic framework for multi-objective optimisation. Our multi-objective hyper-heuristic controls and combines the strengths of three well-known multi-objective evolutionary algorithms (NSGAII, SPEA2, and MOGA), which are utilised as the low level heuristics. A choice function selection heuristic acts as a high level strategy which adaptively ranks the performance of those low-level heuristics according to feedback received during the search process, deciding which one to call at each decision point. Four performance measurements are integrated into a ranking scheme which acts as a feedback learning mechanism to provide knowledge of the problem domain to the high level strategy. To the best of our knowledge, for the first time, this thesis investigates the influence of the move acceptance component of selection hyper-heuristics for multi-objective optimisation. Three multi-objective choice function based hyper-heuristics, combined with different move acceptance strategies including All-Moves as a deterministic move acceptance and the Great Deluge Algorithm (GDA) and Late Acceptance (LA) as a nondeterministic move acceptance function. GDA and LA require a change in the value of a single objective at each step and so a well-known hypervolume metric, referred to as D metric, is proposed for their applicability to the multi-objective optimisation problems. D metric is used as a way of comparing two non-dominated sets with respect to the objective space. The performance of the proposed multi-objective selection choice function based hyper-heuristics is evaluated on the Walking Fish Group (WFG) test suite which is a common benchmark for multi-objective optimisation. Additionally, the proposed approaches are applied to the vehicle crashworthiness design problem, in order to test its effectiveness on a realworld multi-objective problem. The results of both benchmark test problems demonstrate the capability and potential of the multi-objective hyper-heuristic approaches in solving continuous multi-objective optimisation problems. The multi-objective choice function Great Deluge Hyper-Heuristic (HHMO_CF_GDA) turns out to be the best choice for solving these types of problems

Unmet goals of tracking: within-track heterogeneity of students' expectations for

Educational systems are often characterized by some form(s) of ability grouping, like tracking. Although substantial variation in the implementation of these practices exists, it is always the aim to improve teaching efficiency by creating homogeneous groups of students in terms of capabilities and performances as well as expected pathways. If students’ expected pathways (university, graduate school, or working) are in line with the goals of tracking, one might presume that these expectations are rather homogeneous within tracks and heterogeneous between tracks. In Flanders (the northern region of Belgium), the educational system consists of four tracks. Many students start out in the most prestigious, academic track. If they fail to gain the necessary credentials, they move to the less esteemed technical and vocational tracks. Therefore, the educational system has been called a 'cascade system'. We presume that this cascade system creates homogeneous expectations in the academic track, though heterogeneous expectations in the technical and vocational tracks. We use data from the International Study of City Youth (ISCY), gathered during the 2013-2014 school year from 2354 pupils of the tenth grade across 30 secondary schools in the city of Ghent, Flanders. Preliminary results suggest that the technical and vocational tracks show more heterogeneity in student’s expectations than the academic track. If tracking does not fulfill the desired goals in some tracks, tracking practices should be questioned as tracking occurs along social and ethnic lines, causing social inequality