In the sequential resource allocation problem there is a single divisible resource that is divided over a number of clients. Allocations are made in a predetermined order and only upon arrival at a client their demand for the resource is revealed; only the probability distribution of the demand of every client is known to the supplier. We consider this problem from a fairness perspective, where the aim is to balance allocations between individual clients. Several allocation policies have been proposed in the literature. In this work, we introduce a new, non-adaptive policy based on linear programming that can also incorporate group fairness. In addition, we provide an extensive computational study to compare allocation policies on several fairness measures. Using an optimized implementation of existing methods, we are able to evaluate significantly larger problem instances than those previously considered in the literature.

There is arbitrariness in optimum solutions of graph-theoretic problems that can give rise to unfairness. Incorporating fairness in such problems, however, can be done in multiple ways. For instance, fairness can be defined on an individual level, for individual vertices or edges of a given graph, or on a group level. In this work, we analyze in detail two individual-fairness measures that are based on finding a probability distribution over the set of solutions. One measure guarantees uniform fairness, i.e., entities have equal chance of being part of the solution when sampling from this probability distribution. The other measure maximizes the minimum probability for every entity of being selected in a solution. In particular, we reveal that computing these individual-fairness measures is in fact equivalent to computing the fractional covering number and the fractional partitioning number of a hypergraph. In addition, we show that for a general class of problems that we classify as independence systems, these two measures coincide. We also analyze group fairness and how this can be combined with the individual-fairness measures. Finally, we establish the computational complexity of determining group-fair solutions for a variant of the matching problem.

Abstract Set systems can be visualized in various ways. An important distinction between techniques is whether the elements have a spatial location that is to be used for the visualization; for example, the elements are cities on a map. Strictly adhering to such location may severely limit the visualization and force overlay, intersections and other forms of clutter. On the other hand, completely ignoring the spatial dimension omits information and may hide spatial patterns in the data. We study layouts for set systems (or hypergraphs) in which spatial locations are displaced onto concentric circles or a grid, to obtain schematic set visualizations. We investigate the tractability of the underlying algorithmic problems adopting different optimization criteria (e.g. crossings or bends) for the layout structure, also known as the support of the hypergraph. Furthermore, we describe a simulated‐annealing approach to heuristically optimize a combination of such criteria. Using this method in computational experiments, we explore the trade‐offs and dependencies between criteria for computing high‐quality schematic set visualizations.