SAT-Based Approach for Learning Optimal Decision Trees with Non-Binary Features

Decision trees are a popular classification model in machine learning due to their interpretability and performance. Traditionally, decision-tree classifiers are constructed using greedy heuristic algorithms, however these algorithms do not provide guarantees on the quality of the resultant trees. Instead, a recent line of work has studied the use of exact optimization approaches for constructing optimal decision trees. Most of the recent approaches that employ exact optimization are designed for datasets with binary features. While numeric and categorical features can be transformed to binary features, this transformation can introduce a large number of binary features and may not be efficient in practice. In this work, we present a novel SAT-based encoding for decision trees that supports non-binary features and demonstrate how it can be used to solve two well-studied variants of the optimal decision tree problem. We perform an extensive empirical analysis that shows our approach obtains superior performance and is often an order of magnitude faster than the current state-of-the-art exact techniques on non-binary datasets

The evolution of indirect reciprocity under action and assessment generosity

Indirect reciprocity is a mechanism for the evolution of cooperation based on social norms. This mechanism requires that individuals in a population observe and judge each other’s behaviors. Individuals with a good reputation are more likely to receive help from others. Previous work suggests that indirect reciprocity is only effective when all relevant information is reliable and publicly available. Otherwise, individuals may disagree on how to assess others, even if they all apply the same social norm. Such disagreements can lead to a breakdown of cooperation. Here we explore whether the predominantly studied ‘leading eight’ social norms of indirect reciprocity can be made more robust by equipping them with an element of generosity. To this end, we distinguish between two kinds of generosity. According to assessment generosity, individuals occasionally assign a good reputation to group members who would usually be regarded as bad. According to action generosity, individuals occasionally cooperate with group members with whom they would usually defect. Using individual-based simulations, we show that the two kinds of generosity have a very different effect on the resulting reputation dynamics. Assessment generosity tends to add to the overall noise and allows defectors to invade. In contrast, a limited amount of action generosity can be beneficial in a few cases. However, even when action generosity is beneficial, the respective simulations do not result in full cooperation. Our results suggest that while generosity can favor cooperation when individuals use the most simple strategies of reciprocity, it is disadvantageous when individuals use more complex social norms

Bi-Criteria Diverse Plan Selection via Beam Search Approximation

Recent work on diverse planning has focused on a two-step setting where the first step consists of generating a large number of plans, and the second step consists of selecting a subset of plans that maximizes diversity. For the second step, previous work has focused on solving a combinatorial optimization problem for diverse subset selection that can be approximated using greedy search. In this work, we propose a flexible, bi-criteria framework for diverse plan selection. Our framework consists of optimizing both quality and diversity, generalizing previous work and providing flexibility to prioritize one objective over the other. We consider two quality and two diversity measures and show that greedy search guarantees an approximation with a constant ratio for certain configurations based on established results in the literature. To allow users to trade off additional computation for better solutions, we introduce a beam search approximation that generalizes the greedy search, and we provide approximation guarantees on the obtained solutions. Finally, we conduct extensive experiments that show that: (1) our flexible bi-criteria framework allows us to obtain solutions of better quality while still maintaining a high degree of diversity; (2) our beam search approximation obtains significant improvement in performance over greedy search and, for a large number of instances, is able to generate solutions that are equal to or better than those obtained by an exact MIP solver with a significantly higher runtime limit

Neural Sequence Generation with Constraints via Beam Search with Cuts: A Case Study on VRP

In recent years, neural sequence models have been applied successfully to solve combinatorial optimization problems. Solutions, encoded as sequences, are typically generated from trained models via beam search, a search algorithm that generates sequences token-by-token while keeping a fixed number of promising partial solutions at each step. In this paper, we explore the problem of augmenting beam search generation with the enforcement of requirements---hard constraints that any generated solution must adhere to. We propose a hybrid approach, by encoding the requirements in the form of a constraint satisfaction problem (CSP) and iteratively solving the CSP to cut any partial solution within the beam search that is incapable of satisfying the requirements. We study this problem in the context of vehicle routing problems (VRP) further augmented with capacity-related or temporal requirements. We experimentally show that cuts often allow us to satisfy the requirements with negligible impact on solution quality. Without the use of cuts, beam search is shown to be exponentially less likely to satisfy the requirements as the length of the solution increases and/or the requirements are strengthened

Injury burden in individuals aged 50 years or older in the Eastern Mediterranean region, 1990–2019: a systematic analysis from the Global Burden of Disease Study 2019

Background: Injury poses a major threat to health and longevity in adults aged 50 years or older. The increased life expectancy in the Eastern Mediterranean region warrants a further understanding of the ageing population's inevitable changing health demands and challenges. We aimed to examine injury-related morbidity and mortality among adults aged 50 years or older in 22 Eastern Mediterranean countries. Methods: Drawing on data from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019, we categorised the population into adults aged 50–69 years and adults aged 70 years and older. We examined estimates for transport injuries, self-harm injuries, and unintentional injuries for both age groups, with sex differences reported, and analysed the percentage changes from 1990 to 2019. We reported injury-related mortality rates and disability-adjusted life-years (DALYs). The Socio-demographic Index (SDI) and the Healthcare Access and Quality (HAQ) Index were used to better understand the association of socioeconomic factors and health-care system performance, respectively, with injuries and health status in older people. Healthy life expectancy (HALE) was compared with injury-related deaths and DALYs and to the SDI and HAQ Index to understand the effect of injuries on healthy ageing. Finally, risk factors for injury deaths between 1990 and 2019 were assessed. 95% uncertainty intervals (UIs) are given for all estimates. Findings: Estimated injury mortality rates in the Eastern Mediterranean region exceeded the global rates in 2019, with higher injury mortality rates in males than in females for both age groups. Transport injuries were the leading cause of deaths in adults aged 50–69 years (43·0 [95% UI 31·0–51·8] per 100 000 population) and in adults aged 70 years or older (66·2 [52·5–75·5] per 100 000 population), closely followed by conflict and terrorism for both age groups (10·2 [9·3–11·3] deaths per 100 000 population for 50–69 years and 45·7 [41·5–50·3] deaths per 100 000 population for ≥70 years). The highest annual percentage change in mortality rates due to injury was observed in Afghanistan among people aged 70 years or older (400·4% increase; mortality rate 1109·7 [1017·7–1214·7] per 100 000 population). The leading cause of DALYs was transport injuries for people aged 50–69 years (1798·8 [1394·1–2116·0] per 100 000 population) and unintentional injuries for those aged 70 years or older (2013·2 [1682·2–2408·7] per 100 000 population). The estimates for HALE at 50 years and at 70 years in the Eastern Mediterranean region were lower than global estimates. Eastern Mediterranean countries with the lowest SDIs and HAQ Index values had high prevalence of injury DALYs and ranked the lowest for HALE at 50 years of age and HALE at 70 years. The leading injury mortality risk factors were occupational exposure in people aged 50–69 years and low bone mineral density in those aged 70 years or older. Interpretation: Injuries still pose a real threat to people aged 50 years or older living in the Eastern Mediterranean region, mainly due to transport and violence-related injuries. Dedicated efforts should be implemented to devise injury prevention strategies that are appropriate for older adults and cost-effective injury programmes tailored to the needs and resources of local health-care systems, and to curtail injury-associated risk and promote healthy ageing. Funding: Bill & Melinda Gates Foundation