arXiv subjects

Vishwa Prakash HV

Publications and source records attributed to Vishwa Prakash HV.

9 recordsLinked to original sources

arXiv2025

The Landscape of Almost Equitable Allocations

Equitability is a fundamental notion in fair division which requires that all agents derive equal value from their allocated bundles. We study, for general (possibly non-monotone) valuations, a popular relaxation of equitability known as equitability up to one item (EQ1). An EQ1 allocation may fail to exist even with additive non-monotone valuations; for instance, when there are two agents, one valuing every item positively and the other negatively. This motivates a mild and natural assumption: all agents agree on the sign of their value for the grand bundle. Under this assumption, we prove the existence and provide an efficient algorithm for computing EQ1 allocations for two agents with general valuations. When there are more than two agents, we show the existence and polynomial-time computability of EQ1 allocations for valuation classes beyond additivity and monotonicity, in particular for (1) doubly monotone valuations and (2) submodular (resp. supermodular) valuations where the value for the grand bundle is nonnegative (resp. nonpositive) for all agents. Furthermore, we settle an open question of Bil`o et al. by showing that an EQ1 allocation always exists for nonnegative(resp. nonpositive) valuations, i.e., when every agent values each subset of items nonnegatively (resp. nonpositively). Finally, we complete the picture by showing that for general valuations with more than two agents, EQ1 allocations may not exist even when agents agree on the sign of the grand bundle, and that deciding the existence of an EQ1 allocation is computationally intractable.

cs.GT↗

arXiv2025

Fair and Efficient Allocation of Indivisible Mixed Manna

We study fair division of indivisible mixed manna (items whose values may be positive, negative, or zero) among agents with additive valuations. Here, we establish that fairness -- in terms of a relaxation of envy-freeness -- and Pareto efficiency can always be achieved together. Specifically, our fairness guarantees are in terms of envy-freeness up to $k$ reallocations (EFR-$k$): An allocation $A$ of the indivisible items is said to be EFR-$k$ if there exists a subset $R$ of at most $k$ items such that, for each agent $i$, we can reassign items from within $R$ (in $A$) and obtain an allocation, $A^i$, which is envy-free for $i$. We establish that, when allocating mixed manna among $n$ agents with additive valuations, an EFR-$(n-1)$ and Pareto optimal (PO) allocation $A$ always exists. Further, the individual envy-free allocations $A^i$, induced by reassignments, are also PO. In addition, we prove that such fair and efficient allocations are efficiently computable when the number of agents, $n$, is fixed. We also obtain positive results focusing on EFR by itself (and without the PO desideratum). Specifically, we show that an EFR-$(n-1)$ allocation of mixed manna can be computed in polynomial time. In addition, we prove that when all the items are goods, an EFR-${\lfloor n/2 \rfloor}$ allocation exists and can be computed efficiently. Here, the $(n-1)$ bound is tight for chores and $\lfloor n/2 \rfloor$ is tight for goods. Our results advance the understanding of fair and efficient allocation of indivisible mixed manna and rely on a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division. We utilize weighted welfare maximization, with perturbed valuations, to achieve Pareto efficiency, and overall, our techniques are notably different from existing market-based approaches.

cs.GT↗

arXiv2025

Almost and Approximate EFX for Few Types of Agents

We study the problem of fair allocation of a set of indivisible goods among $n$ agents with $k$ distinct additive valuations, with the goal of achieving approximate envy-freeness up to any good ($α-\mathrm{EFX}$). It is known that EFX allocations exist for $n$ agents when there are at most three distinct valuations due to HV et al. Furthermore, Amanatidis et al. showed that a $\frac{2}{3}-\mathrm{EFX}$ allocation is guaranteed to exist when number of agents is at most seven. In this paper, we show that a $\frac{2}{3}-\mathrm{EFX}$ allocation exists for any number of agents when there are at most four distinct valuations. Secondly, we consider a relaxation called $\mathrm{EFX}$ with charity, where some goods remain unallocated such that no agent envies the set of unallocated goods. Akrami et al. showed that for $n$ agents and any $\varepsilon \in \left(0, \frac{1}{2}\right]$, there exists a $(1-\varepsilon)-\mathrm{EFX}$ allocation with at most $\tilde{\mathcal{O}}((n/\varepsilon)^{\frac{1}{2}})$ goods to charity. In this paper, we show that a $(1-\varepsilon)-\mathrm{EFX}$ allocation with a $\tilde{\mathcal{O}}(k/\varepsilon)^{\frac{1}{2}}$ charity exists for any number of agents when there are at most $k$ distinct valuations.

cs.GT↗

arXiv2025

EFX Exists for Three Types of Agents

We study the problem of finding an envy-free allocation of indivisible goods among agents with additive valuations. We focus on the fairness notion of envy-freeness up to any good (EFX). A central open question in fair division is whether EFX allocations always exist for any number of agents. While EFX has been established for three agents [CGM24] and for any number of agents with at most two distinct valuations [Mah23], its existence in more general settings remains open. In this paper, we make significant progress by proving that EFX allocations exist for any number of agents when there are at most three distinct additive valuations. This result simultaneously generalizes both the three-agent case and the two-type case, settling an open question in the field (see [Mah23]).

cs.GT↗

arXiv2025

Best of Both Worlds Guarantees for Equitable Allocations

Equitability is a well-studied fairness notion in fair division, where an allocation is equitable if all agents receive equal utility from their allocation. For indivisible items, an exactly equitable allocation may not exist, and a natural relaxation is EQ1, which stipulates that any inequitability should be resolved by the removal of a single item. In this paper, we study equitability in the context of randomized allocations. Specifically, we aim to achieve equitability in expectation (ex ante EQ) and require that each deterministic outcome in the support satisfies ex post EQ1. Such an allocation is commonly known as a `Best of Both Worlds' allocation, and has been studied, e.g., for envy-freeness and MMS. We characterize the existence of such allocations using a geometric condition on linear combinations of EQ1 allocations, and use this to give comprehensive results on both existence and computation. For two agents, we show that ex ante EQ and ex post EQ1 allocations always exist and can be computed in polynomial time. For three or more agents, however, such allocations may not exist. We prove that deciding existence of such allocations is strongly NP-complete in general, and weakly NP-complete even for three agents. We also present a pseudo-polynomial time algorithm for a constant number of agents. We show that when agents have binary valuations, best of both worlds allocations that additionally satisfy welfare guarantees exist and are efficiently computable.

cs.GT↗

arXiv2025

(Almost Full) EFX for Three (and More) Types of Agents

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.

cs.GT↗

arXiv2024

Fair and Efficient Completion of Indivisible Goods

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.

cs.GT↗

arXiv2023

Fair Healthcare Rationing to Maximize Dynamic Utilities

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

cs.MA↗

arXiv2021

Disjoint Stable Matchings in Linear Time

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 König'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.

cs.DS↗