Allocation of scarce healthcare resources under limited logistic and infrastructural facilities is a major issue in the modern society. We consider the problem of allocation of healthcare resources like vaccines to people or hospital beds to patients in an online manner. Our model takes into account the arrival of resources on a day-to-day basis, different categories of agents, the possible unavailability of agents on certain days, and the utility associated with each allotment as well as its variation over time. We propose a model where priorities for various categories are modelled in terms of utilities of agents. We give online and offline algorithms to compute an allocation that respects eligibility of agents into different categories, and incentivizes agents not to hide their eligibility for some category. The offline algorithm gives an optimal allocation while the online algorithm gives an approximation to the optimal allocation in terms of total utility. Our algorithms are efficient, and maintain fairness among different categories of agents. Our models have applications in other areas like refugee settlement and visa allocation. We evaluate the performance of our algorithms on real-life and synthetic datasets. The experimental results show that the online algorithm is fast and performs better than the given theoretical bound in terms of total utility. Moreover, the experimental results confirm that our utility-based model correctly captures the priorities of categories.

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Pen testing is the problem of selecting high capacity resources when the only way to measure the capacity of a resource expends its capacity. We have a set of n pens with unknown amounts of ink and our goal is to select a feasible subset of pens maximizing the total ink in them. We are allowed to gather more information by writing with them, but this uses up ink that was previously in the pens. Algorithms are evaluated against the standard benchmark, i.e, the optimal pen testing algorithm, and the omniscient benchmark, i.e, the optimal selection if the quantity of ink in the pens are known.We identify optimal and near optimal pen testing algorithms by drawing analogues to auction theoretic frameworks of deferred-acceptance auctions and virtual values. Our framework allows the conversion of any near optimal deferred-acceptance mechanism into a pen testing algorithm with an additional overhead of at most (1 + o(1)) ln n in the approximation factor of the omniscient benchmark. We use this framework to give pen testing algorithms for various combinatorial constraints like matroid, knapsack and general downward-closed constraints and also for online environments.

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