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We formulate the problem of fair and efficient completion of indivisible goods, defined as follows: Given a partial allocation of indivisible goods among agents, does there exist an allocation of the remaining goods (i.e., a completion) that satisfies fairness and economic efficiency guarantees of interest? We study the computational complexity of the completion problem for prominent fairness and efficiency notions such as envy-freeness up one good (EF1), proportionality up to one good (Prop1), maximin share (MMS), and Pareto optimality (PO), and focus on the class of additive valuations as well as its subclasses such as binary additive and lexicographic valuations. We find that while the completion problem is significantly harder than the standard fair division problem (wherein the initial partial allocation is empty), the consideration of restricted preferences facilitates positive algorithmic results for threshold-based fairness notions (Prop1 and MMS). On the other hand, the completion problem remains computationally intractable for envy-based notions such as EF1 and EF1+PO even under restricted preferences., Comment: To appear at the 39th Annual AAAI Conference on Artificial Intelligence (AAAI 2025)

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 on-line 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

We study the problem of determining an envy-free allocation of indivisible goods among multiple agents with additive valuations. EFX, which stands for envy-freeness up to any good, is a well-studied relaxation of the envy-free allocation problem and has been shown to exist for specific scenarios. EFX is known to exist for three agents, and for any number of agents when there are only two types of valuations. EFX allocations are also known to exist for four agents with at most one good unallocated. In this paper, we show that EFX exists with at most k-2 goods unallocated for any number of agents having k distinct valuations. Additionally, we show that complete EFX allocations exist when all but two agents have identical valuations., Comment: To appear at the 39th Annual AAAI Conference on Artificial Intelligence (AAAI 2025)

We show that given a SM instance G as input we can find a largest collection of pairwise edge-disjoint stable matchings of G in time linear in the input size. This extends two classical results: 1. The Gale-Shapley algorithm, which can find at most two ("extreme") pairwise edge-disjoint stable matchings of G in linear time, and 2. The polynomial-time algorithm for finding a largest collection of pairwise edge-disjoint perfect matchings (without the stability requirement) in a bipartite graph, obtained by combining K\"{o}nig's characterization with Tutte's f-factor algorithm. Moreover, we also give an algorithm to enumerate all maximum-length chains of disjoint stable matchings in the lattice of stable matchings of a given instance. This algorithm takes time polynomial in the input size for enumerating each chain. We also derive the expected number of such chains in a random instance of Stable Matching., Comment: Conference: International Workshop on Graph-Theoretic Concepts in Computer Science 2021 (https://wg2021.mimuw.edu.pl)