Abstract $\eta $ -representations are a way of coding sets in computable linear orders that were first introduced by Fellner in his thesis. Limitwise monotonic functions have been used to characterize the sets with $\eta $ -representations, and give characterizations for several variations of $\eta $ -representations. The one exception is the class of sets with strong $\eta $ -representations, the only class where the order type of the representation is unique. We introduce the notion of a connected approximation of a set, a variation on $\Sigma ^0_2$ approximations. We use connected approximations to give a characterization of the many-one degrees of sets with strong $\eta $ -representations as well new characterizations of the variations of $\eta $ -representations with known characterizations.

Dynamic vehicle routing problem with time windows is an important combinatorial optimisation problem in many real-world applications. The most challenging part of the problem is to make real-time decisions (i.e. whether to accept the newly arrived service requests or not) during the execution of the routes. It is hardly applicable to use the optimisation methods such as mathematical programming and evolutionary algorithms that are competitive for static problems, since they are usually time-consuming, and cannot give real-time responses. In this paper, we consider solving this problem using heuristics. A heuristic gradually builds a solution by adding the requests to the end of the route one by one. This way, it can take advantage of the latest information when making the next decision, and give immediate response. In this paper, we propose a meta-algorithm to generate a solution given any heuristic. The meta-algorithm maintains a set of routes throughout the scheduling horizon. Whenever a new request arrives, it tries to re-generate new routes to include the new request by the heuristic. It accepts the new request if successful, and reject otherwise. Then we manually designed several heuristics, and proposed a genetic programming-based hyper-heuristic to automatically evolve heuristics. The results showed that the heuristics evolved by genetic programming significantly outperformed the manually designed heuristics.