Exact Markov Chain-based Runtime Analysis of a Discrete Particle Swarm Optimization Algorithm on Sorting and OneMax

Meta-heuristics are powerful tools for solving optimization problems whose structural properties are unknown or cannot be exploited algorithmically. We propose such a meta-heuristic for a large class of optimization problems over discrete domains based on the particle swarm optimization (PSO) paradigm. We provide a comprehensive formal analysis of the performance of this algorithm on certain "easy" reference problems in a black-box setting, namely the sorting problem and the problem OneMAX. In our analysis we use a Markov-model of the proposed algorithm to obtain upper and lower bounds on its expected optimization time. Our bounds are essentially tight with respect to the Markov-model. We show that for a suitable choice of algorithm parameters the expected optimization time is comparable to that of known algorithms and, furthermore, for other parameter regimes, the algorithm behaves less greedy and more explorative, which can be desirable in practice in order to escape local optima. Our analysis provides a precise insight on the tradeoff between optimization time and exploration. To obtain our results we introduce the notion of indistinguishability of states of a Markov chain and provide bounds on the solution of a recurrence equation with non-constant coefficients by integration

Fairness in examination timetabling: student preferences and extended formulations

Variations of the examination timetabling problem have been investigated by the research community for more than two decades. The common characteristic between all problems is the fact that the definitions and data sets used all originate from actual educational institutions, particularly universities, including specific examination criteria and the students involved. Although much has been achieved and published on the state-of-the-art problem modelling and optimisation, a lack of attention has been focussed on the students involved in the process. This work presents and utilises the results of an extensive survey seeking student preferences with regard to their individual examination timetables, with the aim of producing solutions which satisfy these preferences while still also satisfying all existing benchmark considerations. The study reveals one of the main concerns relates to fairness within the students cohort; i.e. a student considers fairness with respect to the examination timetables of their immediate peers, as highly important. Considerations such as providing an equitable distribution of preparation time between all student cohort examinations, not just a majority, are used to form a measure of fairness. In order to satisfy this requirement, we propose an extension to the state-of-the-art examination timetabling problem models widely used in the scientific literature. Fairness is introduced as a new objective in addition to the standard objectives, creating a multi-objective problem. Several real-world examination data models are extended and the benchmarks for each are used in experimentation to determine the effectiveness of a multi-stage multi-objective approach based on weighted Tchebyceff scalarisation in improving fairness along with the other objectives. The results show that the proposed model and methods allow for the production of high quality timetable solutions while also providing a trade-off between the standard soft constraints and a desired fairness for each student

Operational Research in Education

Operational Research (OR) techniques have been applied, from the early stages of the discipline, to a wide variety of issues in education. At the government level, these include questions of what resources should be allocated to education as a whole and how these should be divided amongst the individual sectors of education and the institutions within the sectors. Another pertinent issue concerns the efficient operation of institutions, how to measure it, and whether resource allocation can be used to incentivise efficiency savings. Local governments, as well as being concerned with issues of resource allocation, may also need to make decisions regarding, for example, the creation and location of new institutions or closure of existing ones, as well as the day-to-day logistics of getting pupils to schools. Issues of concern for managers within schools and colleges include allocating the budgets, scheduling lessons and the assignment of students to courses. This survey provides an overview of the diverse problems faced by government, managers and consumers of education, and the OR techniques which have typically been applied in an effort to improve operations and provide solutions

Directed hypergraph connectivity augmentation by hyperarc reorientations

The orientation theorem of Nash-Williams states that an undirected graph admits a k-arc-connected orientation if and only if it is 2k-edge-connected. Recently, Ito et al. showed that any orientation of an undirected 2k-edge-connected graph can be transformed into a k-arc-connected orientation by reorienting one arc at a time without decreasing the arc-connectivity at any step, thus providing an algorithmic proof of Nash-Williams' theorem. We generalize their result to hypergraphs and therefore provide an algorithmic proof of the characterization of hypergraphs with a k-hyperarc-connected orientation originally given by Frank et al. We prove that any orientation of an undirected (k,k)-partition-connected hypergraph can be transformed into a k-hyperarc-connected orientation by reorienting one hyperarc at a time without decreasing the hyperarc-connectivity in any step. Furthermore, we provide a simple combinatorial algorithm for computing such a transformation in polynomial time

: Approximating network design problems between 1- and 2-connectivity

We introduce and study the problem Flexible Graph Connectivity, which in contrast to many classical connectivity problems features a non-uniform failure model. We distinguish between safe and unsafe resources and postulate that failures can only occur among the unsafe resources. Given an undirected edge-weighted graph and a set of unsafe edges, the task is to find a minimum-cost subgraph that remains connected after removing at most k unsafe edges. We give constant-factor approximation algorithms for this problem for k = 1 as well as for unit costs and k ≥ 1. Our approximation guarantees are close to the known best bounds for special cases, such as the 2-edge-connected spanning subgraph problem and the tree augmentation problem. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min–max–min optimization problem