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We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1, …, mt if and only if n is odd, 3⩽mi⩽n for i=1, …, t, and m1+⋯+mt=MJX-TeXAtom-OPEN(0n2MJX-TeXAtom-CLOSE). We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1, …, mt if and only if n is even, 3⩽mi⩽n for i=1, …, t, and m1+⋯+mt=MJX-TeXAtom-OPEN(0n2MJX-TeXAtom-CLOSE)−n/2.

It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorisation of K into F ; except that there is no factorisation of K 6,6 into F when F is the union of two disjoint 6-cycles.

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We present new integer linear programming (ILP) models for N P-hard optimisation problems in instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals / Residents problem with Ties (HRT). These models can be used to efficiently solve these optimisation problems when applied to (i) instances derived from real-world applications, and (ii) larger instances that are randomlygenerated. In the case of SMTI, we consider instances arising from the pairing of children with adoptive families, where preferences are obtained from a quality measure of each possible pairing of child to family. In this case we seek a maximum weight stable matching. We present new algorithms for preprocessing instances of SMTI with ties on both sides, as well as new ILP models. Algorithms based on existing state-of-the-art models only solve 6 of our 22 real-world instances within an hour per instance, and our new models incorporating dummy variables and constraint merging, together with preprocessing and a warm start, solve all 22 instances within a mean runtime of a minute. For HRT, we consider instances derived from the problem of assigning junior doctors to foundation posts in Scottish hospitals. Here we seek a maximum size stable matching. We show how to extend our models for SMTI to HRT and reduce the average running time for real-world HRT instances by two orders of magnitude. We also show that our models outperform by a wide margin all known state-of-the-art models on larger randomly-generated instances of SMTI and HRT.

The real world applications of optimisation algorithms often are only interested in the running time of an algorithm, which can frequently be significantly reduced through parallelisation. We present two methods of parallelising the recursive algorithm presented by Ozlen, Burton and MacRae [J. Optimization Theory and Applications; 160: [470][471][472][473][474][475][476][477][478][479][480][481][482] 2014]. Both new methods utilise two threads and improve running times. One of the new methods, the Meeting algorithm, halves running time to achieve near-perfect parallelisation, allowing users to solve bi-objective integer problems with more variables.

Spatio-temporal count data relating to a set of non-overlapping areal units are prevalent in many fields, including epidemiology and social science. The spatial autocorrelation inherent in these data is typically modelled by a set of random effects that are assigned a conditional autoregressive prior distribution, which is a special case of a Gaussian Markov random field. The autocorrelation structure implied by this model depends on a binary neighbourhood matrix, where two random effects are assumed to be partially autocorrelated if their areal units share a common border, and are conditionally independent otherwise. This paper proposes a novel graph-based optimisation algorithm for estimating either a static or a temporally varying neighbourhood matrix for the data that better represents its spatial correlation structure, by viewing the areal units as the vertices of a graph and the neighbour relations as the set of edges. The improved estimation performance of our methodology compared to the commonly used border sharing rule is evidenced by simulation, before the method is applied to a new respiratory disease surveillance study in Scotland between 2011 and 2017.

New Exact Approaches Tailored for Kidney Exchange Programs with Hierarchical Objectives Kidney exchange programs increase the rate of living donor kidney transplantation. Whereas effective integer programming models aimed at maximizing the total number of transplants have been proposed in the literature, these cannot always be extended to handle a hierarchy of objectives, which is often a requirement in practice. In “New Algorithms for Hierarchical Optimization in Kidney Exchange Programs,” Delorme, García, Gondzio, Kalcsics, Manlove, and Pettersson introduce a new integer programming framework to solve large-size instances of kidney exchange programs. The authors use an ad hoc preprocessing and a reduced-cost variable fixing algorithm to dramatically decrease the size of the models, and they devise a diving algorithm that exploits the hierarchical structure of the problem to significantly reduce the number of integer programs that need to be solved. They also show that it is possible to transition between models as different layers are traversed in the hierarchy, allowing each layer to be solved with the most effective model. Experiments on three different European kidney exchange programs show that running times can be reduced by up to three orders of magnitude.