Near-linear Time Approximation Schemes for Clustering in Doubling Metrics

We consider the classic Facility Location, k -Median, and k -Means problems in metric spaces of doubling dimension d . We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is Õ(2 (1/ε) O(d2) n) , making a significant improvement over the state-of-the-art algorithms that run in time n (d/ε) O(d) . Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k -Median and k -Means and efficient bicriteria approximation schemes for k -Median with outliers, k -Means with outliers and k -Center.

Complexity Analysis of Generalized and Fractional Hypertree Decompositions

Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H) , its generalized hypertree width ghw(H) , and its fractional hypertree width fhw(H) , respectively. It is known that hw(H)≤ k can be checked in polynomial time for fixed k , while checking ghw(H)≤ k is NP-complete for k ≥ 3 . The complexity of checking fhw(H)≤ k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H)≤ k is NP-complete, even for k=2 . The same construction allows us to prove also the NP-completeness of checking ghw(H)≤ k for k=2 . After that, we identify meaningful restrictions that make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw .

Optimal national prioritization policies for hospital care during the SARS-CoV-2 pandemic

In response to unprecedented surges in the demand for hospital care during the SARS-CoV-2 pandemic, health systems have prioritized patients with COVID-19 to life-saving hospital care to the detriment of other patients. In contrast to these ad hoc policies, we develop a linear programming framework to optimally schedule elective procedures and allocate hospital beds among all planned and emergency patients to minimize years of life lost. Leveraging a large dataset of administrative patient medical records, we apply our framework to the National Health Service in England and show that an extra 50,750–5,891,608 years of life can be gained compared with prioritization policies that reflect those implemented during the pandemic. Notable health gains are observed for neoplasms, diseases of the digestive system, and injuries and poisoning. Our open-source framework provides a computationally efficient approximation of a large-scale discrete optimization problem that can be applied globally to support national-level care prioritization policies.

Participatory Budgeting with Project Groups

We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and---in addition to a global budget limit---there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to being hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.

A novel stochastic simulation approach enables exploration of mechanisms for regulating polarity site movement

Cells polarize their movement or growth toward external directional cues in many different contexts. For example, budding yeast cells grow toward potential mating partners in response to pheromone gradients. Directed growth is controlled by polarity factors that assemble into clusters at the cell membrane. The clusters assemble, disassemble, and move between different regions of the membrane before eventually forming a stable polarity site directed toward the pheromone source. Pathways that regulate clustering have been identified but the molecular mechanisms that regulate cluster mobility are not well understood. To gain insight into the contribution of chemical noise to cluster behavior we simulated clustering within the reaction-diffusion master equation (RDME) framework to account for molecular-level fluctuations. RDME simulations are a computationally efficient approximation, but their results can diverge from the underlying microscopic dynamics. We implemented novel concentration-dependent rate constants that improved the accuracy of RDME-based simulations of cluster behavior, allowing us to efficiently investigate how cluster dynamics might be regulated. Molecular noise was effective in relocating clusters when the clusters contained low numbers of limiting polarity factors, and when Cdc42, the central polarity regulator, exhibited short dwell times at the polarity site. Cluster stabilization occurred when abundances or binding rates were altered to either lengthen dwell times or increase the number of polarity molecules in the cluster. We validated key results using full 3D particle-based simulations. Understanding the mechanisms cells use to regulate the dynamics of polarity clusters should provide insights into how cells dynamically track external directional cues.

Fair colorful k-center clustering

An instance of colorfulk-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius $$\rho $$ ρ such that there exist balls of radius $$\rho $$ ρ around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations: each color/group should receive a similar service guarantee, and second, from the algorithmic challenges it poses: this problem combines the difficulties of clustering along with the subset-sum problem. In particular, we show that this combination results in strong integrality gap lower bounds for several natural linear programming relaxations. Our main result is an efficient approximation algorithm that overcomes these difficulties to achieve an approximation guarantee of 3, nearly matching the tight approximation guarantee of 2 for the classical k-center problem which this problem generalizes. algorithms either opened more than k centers or only worked in the special case when the input points are in the plane.