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Prajakta Nimbhorkar

Publications and source records attributed to Prajakta Nimbhorkar.

At least 19 recordsLinked to original sources
arXiv2026

Nonexistence of Simultaneously EF1 and Pareto Optimal Allocations for Submodular Valuations

The existence of allocations of indivisible goods that are simultaneously fair (envy-free up to one item (EF1)) and efficient (Pareto optimal (PO)) when agents have monotone submodular valuations has been a longstanding open problem. We settle this question negatively by giving an example with two agents where no allocation is simultaneously EF1 and PO. We also show that determining the existence of such allocations is NP-hard for monotone submodular valuations. Our example uses (unweighted) coverage valuations, which is a strict subclass of monotone submodular valuations. Since EF1+PO allocations are known to always exist for additive valuations via the maximization of Nash Social Welfare (Caragiannis et al. (ACM TEAC 2019)), and for matroid-rank valuations (Benabbou et al. (ACM TEAC 2021)), nonexistence was known only for monotone subadditive valuations (Caragiannis et al. (ACM TEAC 2019)). Our work moves the nonexistence frontier to unweighted coverage valuations. We also show that the example we designed for goods also proves nonexistence of EF1+PO in general, for chores with unweighted coverage costs, by interpreting the valuations as disutilities.

cs.GT↗
arXiv2026

Fair Division in a Variable Setting

We study fair division of indivisible items under a variable input setting, where the set of agents or items may change over time. Starting from an arbitrary allocation, the goal is to restore envy-freeness up to one item (EF1) through item transfers while causing as little disruption as possible. We formalize this via `valid transfers' and introduce the EF1-Restoration problem. We give efficient algorithms for EF1-Restoration when agents have identical monotone valuations and the items are either all goods or all chores. In contrast, even for identical additive valuations, we prove that optimizing the number of valid transfers is NP-hard. For the stronger notion of EFX, we show that deciding whether EFX-Restoration admits any positive solution is weakly NP-hard for identical additive valuations. We also show that, unlike the pure goods and pure chores cases, EF1-Restoration may be impossible for mixed manna. For additive binary valuations, we prove that deciding whether EF1-Restoration is possible is NP-hard, and so is finding the minimum number of valid transfers when restoration is possible. We complement these hardness results with a polynomial-time algorithm for the subclass of additive binary valuations defined using multigraphs, introduced by Christodoulou et al. (EC 2023), when allocations are required to be orientations. Finally, for monotone binary valuations, we prove that deciding whether EF1-Restoration is possible is PSPACE-complete. Together, our results give a broad complexity landscape for restoring EF1 under variable inputs across several natural valuation classes.

cs.GT↗
arXiv2026

Optimal Capacity Modification for Stable Matchings with Ties

We consider the Hospitals/Residents (HR) problem in the presence of ties in hospital preferences. Among the three notions of stability, namely weak stability, strong stability, and super-stability, we focus on the notion of strong stability. Strong stability has many desirable properties, both theoretically and in practice; however, its existence is not guaranteed. In this paper, our objective is to optimally increase the quotas of hospitals to ensure that a strongly stable matching exists in the modified instance. We explore two natural optimization criteria: (i) minimizing the total capacity increase across all hospitals (MINSUM) and (ii) minimizing the maximum capacity increase for any hospital (MINMAX). We show that the MINSUM problem admits a polynomial-time algorithm. We also establish an analog of the well-known rural hospitals theorem [Gale & Sotomayor, 1985; Roth, 1986], adapted to the MINSUM augmentation setting. We consider a generalization of the MINSUM problem in which each hospital incurs a cost per unit increase in its quota. We show that the cost version of the MINSUM problem is NP-hard and inapproximable within any multiplicative factor, even if the costs are zero or one. For the MINSUM objective with a set of forced edges, we give a polynomial-time algorithm. In contrast to the above results for the MINSUM problem, we show that the MINMAX problem is NP-hard. When hospital preference lists have ties of length at most $\ell+1$, we give a polynomial-time algorithm that increases each hospital's quota by at most $\ell$, ensuring the resulting instance admits a strongly stable matching. Moreover, among all such augmentations, our algorithm outputs the best strongly stable matching from the residents' perspective.

cs.DS↗
arXiv2026

Critical Relaxed-Stable Matchings with Ties in the Many-to-Many Setting

We study the many-to-many bipartite matching problem in the presence of preferences where ties, as well as lower quotas, may appear on both sides of the bipartition. The input is a bipartite graph $G=(A \cup B, E)$, where each vertex in $A \cup B$ has a positive upper quota and a non-negative lower quota denoting the maximum and minimum number of vertices that can be assigned to it from its neighborhood. Additionally, each vertex specifies a preference ordering, possibly containing ties, over its neighbors. A \textit{critical} matching is a matching which fulfills vertex lower quotas to the maximum possible extent. We seek to compute a matching that is critical as well as optimal with respect to the preferences of vertices. Stability, a well-accepted notion of optimality in the presence of two-sided preferences, is generalized to weak-stability in the presence of ties. However, a matching that is critical as well as weakly stable may not exist. Popularity is another well-investigated notion of optimality for the two-sided preference model; however, in the presence of ties (even without lower quotas), a popular matching may not exist. We, therefore, consider the notion of relaxed stability, which was introduced and studied by Krishnaa, Limaye, Nasre, and Nimbhorkar~(JoCO 2023). We show that a matching that is critical as well as relaxed-stable always exists, although computing a maximum-size relaxed-stable matching turns out to be NP-hard. Our main contribution achieves a $\frac{2}{3}$-approximation to the maximum cardinality critical relaxed-stable matching in polynomial time.

cs.DS↗
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

Group Fair Matchings using Convex Cost Functions

We consider the problem of assigning items to platforms where each item has a utility associated with each of the platforms to which it can be assigned. Each platform has a soft constraint over the total number of items it serves, modeled via a convex cost function. Additionally, items are partitioned into groups, and each platform also incurs group-specific convex cost over the number of items from each group that can be assigned to the platform. These costs promote group fairness by penalizing imbalances, yielding a soft variation of fairness notions introduced in prior work, such as Restricted Dominance and Minority protection. Restricted Dominance enforces upper bounds on group representation, while Minority protection enforces lower bounds. Our approach replaces such hard constraints with cost-based penalties, allowing more flexible trade-offs. Our model also captures Nash Social Welfare kind of objective. The cost of an assignment is the sum of the values of all the cost functions across all the groups and platforms. The objective is to find an assignment that minimizes the cost while achieving a total utility that is at least a user-specified threshold. The main challenge lies in balancing the overall platform cost with group-specific costs, both governed by convex functions, while meeting the utility constraint. We present an efficient polynomial-time approximation algorithm, supported by theoretical guarantees and experimental evaluation. Our algorithm is based on techniques involving linear programming and network flows. We also provide an exact algorithm for a special case with uniform utilities and establish the hardness of the general problem when the groups can intersect arbitrarily.

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

(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

Individual Fairness under Varied Notions of Group Fairness in Bipartite Matching - One Framework to Approximate Them All

We study the probabilistic assignment of items to platforms that satisfies both group and individual fairness constraints. Each item belongs to specific groups and has a preference ordering over platforms. Each platform enforces group fairness by limiting the number of items per group that can be assigned to it. There could be multiple optimal solutions that satisfy the group fairness constraints, but this alone ignores item preferences. Our approach explores a `best of both worlds fairness' solution to get a randomized matching, which is ex-ante individually fair and ex-post group-fair. Thus, we seek a `probabilistic individually fair' distribution over `group-fair' matchings where each item has a `high' probability of matching to one of its top choices. This distribution is also ex-ante group-fair. Users can customize fairness constraints to suit their requirements. Our first result is a polynomial-time algorithm that computes a distribution over `group-fair' matchings such that the individual fairness constraints are approximately satisfied and the expected size of a matching is close to OPT. We empirically test this on real-world datasets. We present two additional polynomial-time bi-criteria approximation algorithms that users can choose from to balance group fairness and individual fairness trade-offs. For disjoint groups, we provide an exact polynomial-time algorithm adaptable to additional lower `group fairness' bounds. Extending our model, we encompass `maxmin group fairness,' amplifying underrepresented groups, and `mindom group fairness,' reducing the representation of dominant groups.'

cs.AI↗
arXiv2023

Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings

We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.

cs.GT↗
arXiv2023

Online Algorithms for Matchings with Proportional Fairness Constraints and Diversity Constraints

Matching problems with group-fairness constraints and diversity constraints have numerous applications such as in allocation problems, committee selection, school choice, etc. Moreover, online matching problems have lots of applications in ad allocations and other e-commerce problems like product recommendation in digital marketing. We study two problems involving assigning {\em items} to {\em platforms}, where items belong to various {\em groups} depending on their attributes; the set of items are available offline and the platforms arrive online. In the first problem, we study online matchings with {\em proportional fairness constraints}. Here, each platform on arrival should either be assigned a set of items in which the fraction of items from each group is within specified bounds or be assigned no items; the goal is to assign items to platforms in order to maximize the number of items assigned to platforms. In the second problem, we study online matchings with {\em diversity constraints}, i.e. for each platform, absolute lower bounds are specified for each group. Each platform on arrival should either be assigned a set of items that satisfy these bounds or be assigned no items; the goal is to maximize the set of platforms that get matched. We study approximation algorithms and hardness results for these problems. The technical core of our proofs is a new connection between these problems and the problem of matchings in hypergraphs. Our experimental evaluation shows the performance of our algorithms on real-world and synthetic datasets exceeds our theoretical guarantees.

cs.DS↗
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↗
arXiv2023

Popular Critical Matchings in the Many-to-Many Setting

We consider the many-to-many bipartite matching problem in the presence of two-sided preferences and two-sided lower quotas. The input to our problem is a bipartite graph G=(A U B, E), where each vertex in A U B specifies a strict preference ordering over its neighbors. Each vertex has an upper quota and a lower quota denoting the maximum and minimum number of vertices that can be assigned to it from its neighborhood. In the many-to-many setting with two-sided lower quotas, informally, a critical matching is a matching which fulfils vertex lower quotas to the maximum possible extent. This is a natural generalization of the definition of critical matching in the one-to-one setting [Kavitha T., FSTTCS 2021]. Our goal in the given problem is to find a popular matching in the set of critical matchings. A matching is popular in a given set of matchings if it remains undefeated in a head-to-head election with any matching in that set. Here, vertices cast votes between pairs of matchings. We show that there always exists a matching that is popular in the set of critical matchings. We present an efficient algorithm to compute such a matching of the largest size. We prove the popularity of our matching using a dual certificate.

cs.DS↗
arXiv2022

Popular Edges with Critical Nodes

In the popular edge problem, the input is a bipartite graph $G = (A \cup B,E)$ where $A$ and $B$ denote a set of men and a set of women respectively, and each vertex in $A\cup B$ has a strict preference ordering over its neighbours. A matching $M$ in $G$ is said to be {\em popular} if there is no other matching $M'$ such that the number of vertices that prefer $M'$ to $M$ is more than the number of vertices that prefer $M$ to $M'$. The goal is to determine, whether a given edge $e$ belongs to some popular matching in $G$. A polynomial-time algorithm for this problem appears in \cite{CK18}. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from \cite{Kavitha14,Kavitha21,NNRS21,NN17} that, when $G$ admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.

cs.DS↗
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↗
arXiv2021

Matchings with Group Fairness Constraints: Online and Offline Algorithms

We consider the problem of assigning items to platforms in the presence of group fairness constraints. In the input, each item belongs to certain categories, called classes in this paper. Each platform specifies the group fairness constraints through an upper bound on the number of items it can serve from each class. Additionally, each platform also has an upper bound on the total number of items it can serve. The goal is to assign items to platforms so as to maximize the number of items assigned while satisfying the upper bounds of each class. In some cases, there is a revenue associated with matching an item to a platform, then the goal is to maximize the revenue generated. This problem models several important real-world problems like ad-auctions, scheduling, resource allocations, school choice etc.We also show an interesting connection to computing a generalized maximum independent set on hypergraphs and ranking items under group fairness constraints. We show that if the classes are arbitrary, then the problem is NP-hard and has a strong inapproximability. We consider the problem in both online and offline settings under natural restrictions on the classes. Under these restrictions, the problem continues to remain NP-hard but admits approximation algorithms with small approximation factors. We also implement some of the algorithms. Our experiments show that the algorithms work well in practice both in terms of efficiency and the number of items that get assigned to some platform.

cs.DS↗
arXiv2020

Envy-freeness and Relaxed Stability under lower quotas

We consider the problem of matchings under two-sided preferences in the presence of maximum as well as minimum quota requirements for the agents. This setting, studied as the Hospital Residents with Lower Quotas (HRLQ) in literature, models important real world problems like assigning medical interns (residents) to hospitals, and teaching assistants to instructors where a minimum guarantee is essential. When there are no minimum quotas, stability is the de-facto notion of optimality. However, in the presence of minimum quotas, ensuring stability and simultaneously satisfying lower quotas is not an attainable goal in many instances. To address this, a relaxation of stability known as envy-freeness, is proposed in literature. In our work, we thoroughly investigate envy-freeness from a computational view point. Our results show that computing envy-free matchings that match maximum number of agents is computationally hard and also hard to approximate up to a constant factor. Additionally, it is known that envy-free matchings satisfying lower-quotas may not exist. To circumvent these drawbacks, we propose a new notion called relaxed stability. We show that relaxed stable matchings are guaranteed to exist even in the presence of lower-quotas. Despite the computational intractability of finding a largest matching that is feasible and relaxed stable, we give efficient algorithms that compute a constant factor approximation to this matching in terms of size.

cs.DS↗
arXiv2018

Dichotomy Results for Classified Rank-Maximal Matchings and Popular Matchings

In this paper, we consider the problem of computing an optimal matching in a bipartite graph where elements of one side of the bipartition specify preferences over the other side, and one or both sides can have capacities and classifications. The input instance is a bipartite graph G=(A U P,E), where A is a set of applicants, P is a set of posts, and each applicant ranks its neighbors in an order of preference, possibly involving ties. Moreover, each vertex v in A U P has a quota q(v) denoting the maximum number of partners it can have in any allocation of applicants to posts - referred to as a {\em matching} in this paper. A classification for a vertex u is a collection of subsets of neighbors of u. Each subset (class) has an upper quota denoting the maximum number of vertices from the class that can be matched to u. The goal is to find a matching that is optimal amongst all the feasible matchings, which are matchings that respect quotas of all the vertices and classes. We consider two well-studied notions of optimality namely popularity and rank-maximality. The notion of rank-maximality involves finding a matching in $G$ with maximum number of rank-$1$ edges, subject to that, maximum number of rank-2 edges and so on. We present an O(|E|^2)-time algorithm for finding a feasible rank-maximal matching, when each classification is a laminar family. We complement this with an NP-hardness result when classes are non-laminar even under strict preference lists, and even when only posts have classifications, and each applicant has a quota of one. We show an analogous dichotomy result for computing a popular matching amongst feasible matchings (if one exists) in a bipartite graph with posts having capacities and classifications and applicants having a quota of one.

cs.DS↗