Precedence-Constrained Arborescences

The minimum-cost arborescence problem is a well-studied problem in the area of graph theory, with known polynomial-time algorithms for solving it. Previous literature introduced new variations on the original problem with different objective function and/or constraints. Recently, the Precedence-Constrained Minimum-Cost Arborescence problem was proposed, in which precedence constraints are enforced on pairs of vertices. These constraints prevent the formation of directed paths that violate precedence relationships along the tree. We show that this problem is NP-hard, and we introduce a new scalable mixed integer linear programming model for it. With respect to the previous models, the newly proposed model performs substantially better. This work also introduces a new variation on the minimum-cost arborescence problem with precedence constraints. We show that this new variation is also NP-hard, and we propose several mixed integer linear programming models for formulating the problem

Monte Carlo sampling for the tourist trip design problem

Introduction: The Tourist Trip Design Problem is a variant of a route-planning problem for tourists interested in multiple points of interest. Each point of interest has different availability, and a certain satisfaction score can be achieved when it is visited. Objectives: The objective is to select a subset of points of interests to visit within a given time budget, in such a way that the satisfaction score of the tourist is maximized and the total travel time is minimized. Methods: In our proposed model, the calculation of the availability of a POI is based on the waiting time and / or the weather forecast. However, research shows that most tourists prefer to travel within a crowded and limited area of very attractive POIs for safety reasons and because they feel more in control. Results: In this work we demonstrate that the existing model of the Probabilistic Orienteering Problem fits a probabilistic variant of this problem and that Monte Carlo Sampling techniques can be used inside a heurist solver to efficiently provide solutions. Conclusions: In this work we demonstrate the existing model of the Probabilistic Orienteering Problem fits the stochastic Tourist Trip Design Problem. We proposed a way to solve the problem by using Monte Carlo Sampling techniques inside a heuristic solver and discussed several possible improvements on the model. Further extension of the model will be developed for solving more practical problems.info:eu-repo/semantics/publishedVersio

Advanced metaheuristics for the probabilistic orienteering problem

Stochastic Optimization Problems take uncertainty into account. For this reason they are in general more realistic than deterministic ones, meanwhile, more difficult to solve. The challenge is both on modelling and computation aspects: exact methods usually work only for small instances, besides, there are several problems with no closed-form expression or hard- to-compute objective functions. A state-of-the-art approach for several stochastic/probabilistic vehicle routing problems is to approximate their cost using Monte Carlo sampling. The Orienteering Problem is a routing problem aiming at selecting a subset of a given set of customers to be visited within a given time budget, so that a total revenue is maximized. Multiple stochastic variants of the problem have been studied. The Probabilistic Orienteering Problem is one of these variants, where customers will require a visit according to a certain given probability. The objective is to select a subset of customers to visit within a given time budget, so that an expected total reward is maximized while the expected travel time is minimised. The problem is NP-hard. In this work we propose different metaheuristics based on hybrid Monte Carlo sampling approximation to solve the problem. Detailed computational studies are presented, with the aim of studying the performance of the metaheuristics in terms of precision and speed, while positioning the new method within the existing literature. In this work, we also study the use of Machine Learning tools to help solve optimization problems. By shifting the problem of selecting the number of samples used by the Monte Carlo approximation to that of choosing a trade off between speed and precision, the best number of samples can be predicted by using Machine Learning models in a fast and efficient way. The Tourist Trip Design Problem (TTDP) is a variant of a route-planning problem for tourists interested in visiting multiple points of interest. A practical application of the POP to the probabilistic version of the TTDP is also discussed, and this provides inspiration for more possible applications

Maximum Independent Sets and Supervised Learning

The paper discusses an enhancement to a recently presented supervised learning algorithm to solve the Maximum Independent Set problem. In particular, it is shown that the algorithm can be improved by simplifying the task learnt by the neural network adopted, with measurable effects on the quality of the solutions provided on unseen instances. Empirical results are presented to validate the idea

Monte Carlo Sampling for the Probabilistic Orienteering Problem

The Probabilistic Orienteering Problem is a variant of the orienteering problem where customers are available with a certain probability. Given a solution, the calculation of the objective function value is complex since there is no linear expression for the expected total cost. In this work we approximate the objective function value with a Monte Carlo Sampling technique and present a computational study about precision and speed of such a method. We show that the evaluation based on Monte Carlo Sampling is fast and suitable to be used inside heuristic solvers. Monte Carlo Sampling is also used as a decisional tool to heuristically understand how many of the customers of a tour can be effectively visited before the given deadline is incurred

Notice of Removal: Reinforcement Learning and Additional Rewardsfor the Traveling Salesman Problem

A comprehensive literature on the Traveling Salesman Problem (TSP) is available, and this problem has become a valuable benchmark to test new heuristic methods for general Combinatorial Optimisation problems. For this reason, recently developed Deep Learning-driven heuristics have been tried on the TSP. These Deep Learning frameworks use the city coordinates as inputs, and are trained using reinforcement learning to predict a distribution over the TSP feasible solutions. The aim of the present work is to show how easy-to-calculate Combinatorial Optimization concepts can improve the performances of such systems. In particular, we show how passing Minimum Spanning Tree information during training can lead to significant improvements to the quality of TSP solutions. As a side result, we also propose a Deep Learning architecture able to predict in real time the optimal length of a TSP instance. The proposed architectures have been tested on random 2D Euclidean graphs with 50 and 100 nodes, showing significant results