A 3/2-Approximation for the Metric Many-visits Path TSP

In the Many-visits Path TSP, we are given a set of $n$ cities along with their pairwise distances (or cost) $c(uv)$, and moreover each city $v$ comes with an associated positive integer request $r(v)$. The goal is to find a minimum-cost path, starting at city $s$ and ending at city $t$, that visits each city $v$ exactly $r(v)$ times. We present a $\frac32$-approximation algorithm for the metric Many-visits Path TSP, that runs in time polynomial in $n$ and poly-logarithmic in the requests $r(v)$. Our algorithm can be seen as a far-reaching generalization of the $\frac32$-approximation algorithm for Path TSP by Zenklusen (SODA 2019), which answered a long-standing open problem by providing an efficient algorithm which matches the approximation guarantee of Christofides' algorithm from 1976 for metric TSP. One of the key components of our approach is a polynomial-time algorithm to compute a connected, degree bounded multigraph of minimum cost. We tackle this problem by generalizing a fundamental result of Kir\'aly, Lau and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for general polymatroids, even allowing element multiplicities. Our result directly yields a $\frac32$-approximation to the metric Many-visits TSP, as well as a $\frac32$-approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.Comment: arXiv admin note: text overlap with arXiv:1911.0989

A time- and space-optimal algorithm for the many-visits TSP

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of $n$ cities that visits each city $c$ a prescribed number $k_c$ of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time $n^{O(n)} + O(n^3 \log \sum_c k_c )$ and requires $n^{\Theta(n)}$ space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length $\sum_c k_c$ of the tour, allowing the algorithm to handle instances with very long tours. The \emph{superexponential} dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time $2^{O(n)}$, i.e.\ \emph{single-exponential} in the number of cities, using \emph{polynomial} space. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.Comment: Small fixes, journal versio

Degree-bounded generalized polymatroids and approximating the metric many-visits TSP

In the Bounded Degree Matroid Basis Problem, we are given a matroid and a hypergraph on the same ground set, together with costs for the elements of that set as well as lower and upper bounds $f(\varepsilon)$ and $g(\varepsilon)$ for each hyperedge $\varepsilon$. The objective is to find a minimum-cost basis $B$ such that $f(\varepsilon) \leq |B \cap \varepsilon| \leq g(\varepsilon)$ for each hyperedge $\varepsilon$. Kir\'aly et al. (Combinatorica, 2012) provided an algorithm that finds a basis of cost at most the optimum value which violates the lower and upper bounds by at most $2 \Delta-1$, where $\Delta$ is the maximum degree of the hypergraph. When only lower or only upper bounds are present for each hyperedge, this additive error is decreased to $\Delta-1$. We consider an extension of the matroid basis problem to generalized polymatroids, or g-polymatroids, and additionally allow element multiplicities. The Bounded Degree g-polymatroid Element Problem with Multiplicities takes as input a g-polymatroid $Q(p,b)$ instead of a matroid, and besides the lower and upper bounds, each hyperedge $\varepsilon$ has element multiplicities $m_\varepsilon$. Building on the approach of Kir\'aly et al., we provide an algorithm for finding a solution of cost at most the optimum value, having the same additive approximation guarantee. As an application, we develop a $1.5$-approximation for the metric Many-Visits TSP, where the goal is to find a minimum-cost tour that visits each city $v$ a positive $r(v)$ number of times. Our approach combines our algorithm for the Bounded Degree g-polymatroid Element Problem with Multiplicities with the principle of Christofides' algorithm from 1976 for the (single-visit) metric TSP, whose approximation guarantee it matches.Comment: 17 page

SciSports: Learning football kinematics through two-dimensional tracking data

SciSports is a Dutch startup company specializing in football analytics. This paper describes a joint research effort with SciSports, during the Study Group Mathematics with Industry 2018 at Eindhoven, the Netherlands. The main challenge that we addressed was to automatically process empirical football players' trajectories, in order to extract useful information from them. The data provided to us was two-dimensional positional data during entire matches. We developed methods based on Newtonian mechanics and the Kalman filter, Generative Adversarial Nets and Variational Autoencoders. In addition, we trained a discriminator network to recognize and discern different movement patterns of players. The Kalman-filter approach yields an interpretable model, in which a small number of player-dependent parameters can be fit; in theory this could be used to distinguish among players. The Generative-Adversarial-Nets approach appears promising in theory, and some initial tests showed an improvement with respect to the baseline, but the limits in time and computational power meant that we could not fully explore it. We also trained a Discriminator network to distinguish between two players based on their trajectories; after training, the network managed to distinguish between some pairs of players, but not between others. After training, the Variational Autoencoders generated trajectories that are difficult to distinguish, visually, from the data. These experiments provide an indication that deep generative models can learn the underlying structure and statistics of football players' trajectories. This can serve as a starting point for determining player qualities based on such trajectory data.Comment: This report was made for the Study Group Mathematics with Industry 201

EASY-APP : An artificial intelligence model and application for early and easy prediction of severity in acute pancreatitis

Acute pancreatitis (AP) is a potentially severe or even fatal inflammation of the pancreas. Early identification of patients at high risk for developing a severe course of the disease is crucial for preventing organ failure and death. Most of the former predictive scores require many parameters or at least 24 h to predict the severity; therefore, the early therapeutic window is often missed.The early achievable severity index (EASY) is a multicentre, multinational, prospective and observational study (ISRCTN10525246). The predictions were made using machine learning models. We used the scikit-learn, xgboost and catboost Python packages for modelling. We evaluated our models using fourfold cross-validation, and the receiver operating characteristic (ROC) curve, the area under the ROC curve (AUC), and accuracy metrics were calculated on the union of the test sets of the cross-validation. The most critical factors and their contribution to the prediction were identified using a modern tool of explainable artificial intelligence called SHapley Additive exPlanations (SHAP).The prediction model was based on an international cohort of 1184 patients and a validation cohort of 3543 patients. The best performing model was an XGBoost classifier with an average AUC score of 0.81 ± 0.033 and an accuracy of 89.1%, and the model improved with experience. The six most influential features were the respiratory rate, body temperature, abdominal muscular reflex, gender, age and glucose level. Using the XGBoost machine learning algorithm for prediction, the SHAP values for the explanation and the bootstrapping method to estimate confidence, we developed a free and easy-to-use web application in the Streamlit Python-based framework (http://easy-app.org/).The EASY prediction score is a practical tool for identifying patients at high risk for severe AP within hours of hospital admission. The web application is available for clinicians and contributes to the improvement of the model

Early prediction of acute necrotizing pancreatitis by artificial intelligence : a prospective cohort-analysis of 2387 cases

Pancreatic necrosis is a consistent prognostic factor in acute pancreatitis (AP). However, the clinical scores currently in use are either too complicated or require data that are unavailable on admission or lack sufficient predictive value. We therefore aimed to develop a tool to aid in necrosis prediction. The XGBoost machine learning algorithm processed data from 2387 patients with AP. The confidence of the model was estimated by a bootstrapping method and interpreted via the 10th and the 90th percentiles of the prediction scores. Shapley Additive exPlanations (SHAP) values were calculated to quantify the contribution of each variable provided. Finally, the model was implemented as an online application using the Streamlit Python-based framework. The XGBoost classifier provided an AUC value of 0.757. Glucose, C-reactive protein, alkaline phosphatase, gender and total white blood cell count have the most impact on prediction based on the SHAP values. The relationship between the size of the training dataset and model performance shows that prediction performance can be improved. This study combines necrosis prediction and artificial intelligence. The predictive potential of this model is comparable to the current clinical scoring systems and has several advantages over them