arXiv subjects

Paritosh Verma

Publications and source records attributed to Paritosh Verma.

At least 19 recordsLinked to original sources

arXiv2026

Fair Division Under Boolean Valuations: Beyond Normalization

We study fair division of indivisible items when agents have arbitrary two-level preferences: the value of each agent for any set of items is Boolean, which need not be monotone or additive. Notably, we do not impose the standard assumption of normalization, i.e., different agents may value the empty set at different Boolean levels. Since the preferences are nonmonotone, envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) each admit several variants, depending on which items are tested for removal and whether they are removed from the envious agent's bundle or the envied agent's bundle. This paper investigates the existence of these variants of EF1 and EFX, on their own and together with economic efficiency, incentive compatibility, feasibility constraints, and lottery-based randomization. Our results highlight that the existence landscape depends crucially on the number of normalized agents, who value the empty bundle at the lower Boolean level. The authors used significant assistance from GPT-5.6-Sol for deriving theoretical results, verified any AI-generated proofs for correctness, and expanded on the exposition and simplified arguments, with the aid of GPT-5.6-Sol and Claude Opus 5.

cs.GT↗

arXiv2026

Fair and Efficient Balanced Allocations for Additive Valuations

We study the existence of fair and efficient allocations of indivisible goods under the balancedness constraint, which requires that any two agents' bundles differ in size by at most one. Our main result establishes the existence of balanced allocations that satisfy envy-freeness up to one good (EF1) and fractional Pareto optimality (fPO) for arbitrary additive valuations. This generalizes a recent result of Kawase et al. (2026), which establishes existence only for personalized bivalued valuations or when there are at most two distinct valuation types. Our proof applies the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a weighted-welfare duality framework and develops a novel price-interlacing lemma to overcome barriers encountered by prior work. We extend this technique to category constraints, also known as partition-matroid constraints. In this setting, we establish the existence of an fPO allocation satisfying a weaker, category-sensitive relaxation of EF1, under which envy can be eliminated by removing at most one good from each category. All proofs in this paper were obtained using GPT-5.6-Sol with guidance from the authors. The authors verified the proofs, expanded the exposition, and simplified the arguments with assistance from GPT-5.6-Sol and Claude Fable 5.

cs.GT↗

arXiv2026

Fair Division Under Inaccurate Preferences

The fair allocation of scarce resources is a central problem in mathematics, computer science, operations research, and economics. While much of the fair-division literature assumes that individuals have underlying cardinal preferences, eliciting exact numerical values is often cognitively burdensome and prone to inaccuracies. A growing body of work in fair division addresses this challenge by assuming access only to ordinal preferences. However, the restricted expressiveness of ordinal preferences makes it challenging to quantify and optimize cardinal fairness objectives such as envy. In this paper, we explore the broad landscape of fair division of indivisible items given inaccurate cardinal preferences, with a focus on minimizing envy. We consider various settings based on whether the true preferences of the agents are stochastic or worst-case, and whether the inaccuracies, modeled as additive noise, are stochastic or worst-case. When the true preferences are stochastic, we show that envy-free allocations can be computed with high probability; this is achieved both in the setting with stochastic and worst-case noise. This generalizes a notable result in stochastic fair division, which establishes a similar guarantee, albeit in the absence of any noise. When the true preferences are worst-case, and the noise is bounded, we analyze the maximum envy achieved by the Round-Robin algorithm. This bound is shown to be tight for deterministic algorithms, and applications of this bound are provided. Lastly, we consider a setting with worst-case preferences and noise, where the true preferences for each item are revealed upon its allocation. Here, we give an efficient online algorithm that guarantees logarithmic maximum envy with high probability. This result generalizes a known result from algorithmic discrepancy to a setting with noisy input data.

cs.GT↗

arXiv2025

Multiple mountains on a pulsar: implications for gravitational waves and the spin-down rate

A pulsar, i.e., a spinning neutron star, with a deformation could emit gravitational waves continuously. Such continuous waves, which have not been detected yet, will be very useful to study gravitational physics and to probe the extreme physics of neutron stars. While typically such waves from a pulsar are estimated considering an overall stellar ellipticity, there can be multiple irregularities or mountains in the stellar crust that the gravity of the star cannot smooth. In this paper, we consider this realistic situation and compute the strain, power, torque and the pulsar spin-down rate due to multiple mountains supported by the stellar crust. Here, we consider astronomically motivated mountain distributions and use the Brans-Dicke theory of gravity which has three polarization states: two tensors dominated by the time-varying quadrupole moment and one scalar dominated by the time-varying dipole moment. We also give the limiting results for general relativity.

gr-qc↗

arXiv2025

Introspectively Envy-Free and Efficient Allocation of Indivisible Mixed Manna

The existence of allocations that are fair and efficient, simultaneously, is a central inquiry in fair division literature. A prominent result in discrete fair division shows that the complementary desiderata of fairness and efficiency can be achieved together when allocating indivisible items with nonnegative values; specifically, for indivisible goods and among agents with additive valuations, there always exists an allocation that is both envy-free up to one item (EF1) and Pareto efficient (PO). While a recent breakthrough extends the EF1 and PO guarantee to indivisible chores (items with negative values), the question remains open for indivisible mixed manna, i.e., for indivisible items whose values can be positive, negative, or zero. The current work makes notable progress in resolving this central question. For indivisible mixed manna and additive valuations, we establish the existence of allocations that are PO and introspectively envy-free up to one item (IEF1). In an IEF1 allocation, each agent can eliminate its envy towards all the other agents by either adding an item or removing an item from its own bundle. The notion of IEF1 coincides with EF1 for indivisible chores, and hence, our result generalizes the aforementioned existence guarantee for chores. Our techniques can be adopted to obtain an alternative proof for the existence of EF1 and PO allocations of indivisible goods. Hence, along with the result for mixed manna, we provide a unified approach for establishing the EF1 and PO guarantee for indivisible goods and indivisible chores. We also utilize our result for indivisible items to develop a distinct proof of the noted EF and PO guarantee for divisible mixed manna. Our work highlights an interesting application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division and develops multiple, novel structural insights and algorithmic ideas.

cs.GT↗

arXiv2025

Influence of dark photon on magnetized and charged particle orbits around static spherically symmetric black hole

We elaborate the problem of magnetized particle motion in the spacetime of a static, spherically symmetric black hole influenced by weak magnetic fields stemming from visible and dark matter sectors. The Wald's procedure for obtaining the weakly magnetized solution, generalized to the case of dark photon - Einstein-Maxwell gravity was implemented. The collision process analysis of two particles in the background of the black hole has been studied in order to find the signature of dark matter presence in the nearby of the object in question.

gr-qc↗

arXiv2025

Can synchrotron radiation reveal the presence of {\it dark sector} around black hole?

We studied synchrotron radiation of a massive charged under visible and hidden sector groups, moving in equatorial plane around spherically symmetric weakly magnetized black hole. As a model of dark matter we choose the one, in which Maxwell field is coupled to the additional $U(1)$-gauge field envisaging the dark sector, the so-called dark photon model. Magnetization of a black hole also stems from Maxwell-dark photon electrodynamics. One found the radiation power and energy loss of the particle and looked for the imprints of dark matter on those phenomena.

gr-qc↗

arXiv2025

Online Envy Minimization and Multicolor Discrepancy: Equivalences and Separations

We consider the fundamental problem of allocating $T$ indivisible items that arrive over time to $n$ agents with additive preferences, with the goal of minimizing envy. This problem is tightly connected to online multicolor discrepancy: vectors $v_1, \dots, v_T \in \mathbb{R}^d$ with $\| v_i \|_2 \leq 1$ arrive over time and must be, immediately and irrevocably, assigned to one of $n$ colors to minimize $\max_{i,j \in [n]} \| \sum_{v \in S_i} v - \sum_{v \in S_j} v \|_{\infty}$ at each step, where $S_\ell$ is the set of vectors that are assigned color $\ell$. The special case of $n = 2$ is called online vector balancing. Any bound for multicolor discrepancy implies the same bound for envy minimization. Against an adaptive adversary, both problems have the same optimal bound, $\Theta(\sqrt{T})$, but whether this holds for weaker adversaries is unknown. Against an oblivious adversary, Alweiss et al. give a $O(\log T)$ bound, with high probability, for multicolor discrepancy. Kulkarni et al. improve this to $O(\sqrt{\log T})$ for vector balancing and give a matching lower bound. Whether a $O(\sqrt{\log T})$ bound holds for multicolor discrepancy remains open. These results imply the best-known upper bounds for envy minimization (for an oblivious adversary) for $n$ and two agents, respectively; whether better bounds exist is open. In this paper, we resolve all aforementioned open problems. We prove that online envy minimization and multicolor discrepancy are equivalent against an oblivious adversary: we give a $O(\sqrt{\log T})$ upper bound for multicolor discrepancy, and a $\Omega(\sqrt{\log T})$ lower bound for envy minimization. For a weaker, i.i.d. adversary, we prove a separation: For online vector balancing, we give a $\Omega\left(\sqrt{\frac{\log T}{\log \log T}}\right)$ lower bound, while for envy minimization, we give an algorithm that guarantees a constant upper bound.

cs.GT↗

arXiv2025

Fair Division Beyond Monotone Valuations with Applications to Equitable Graph Partitioning

This paper studies fair division of divisible and indivisible items among agents whose cardinal preferences are not necessarily monotone. We establish the existence of fair divisions and develop approximation algorithms to compute them. We address two complementary valuation classes, subadditive and nonnegative, which go beyond monotone functions. Considering both the division of cake (divisible resources) and allocation of indivisible items, we obtain fairness guarantees in terms of (approximate) envy-freeness (EF) and equability (EQ). In the context of envy-freeness, we prove that an EF division of a cake always exists under cake valuations that are subadditive and globally nonnegative. This result complements the nonexistence of EF allocations for burnt cakes known for more general valuations. In the indivisible-items setting, we establish the existence of EFE3 allocations for subadditive and globally nonnegative valuations. In addition, we obtain universal existence of EFE3 allocations under nonnegative valuations. We study equitability under nonnegative valuations. Here, we prove that EQE3 allocations always exist when the agents' valuations are nonnegative. Also, in the indivisible-items setting, we develop an approximation algorithm that, for given nonnegative valuations, finds allocations that are equitable within additive margins. Our results have combinatorial implications. For instance, the developed results imply the universal existence of proximately dense subgraphs: Given any graph $G=(V, E)$ and integer $k$ (at most $|V|$), there always exists a partition $V_1, V_2, \ldots, V_k$ of the vertex set such that the edge densities within the parts, $V_i$, are additively within four of each other. Further, such a partition can be computed efficiently.

cs.GT↗

arXiv2024

Automating Food Drop: The Power of Two Choices for Dynamic and Fair Food Allocation

Food waste and food insecurity are two closely related pressing global issues. Food rescue organizations worldwide run programs aimed at addressing the two problems. In this paper, we partner with a non-profit organization in the state of Indiana that leads \emph{Food Drop}, a program that is designed to redirect rejected truckloads of food away from landfills and into food banks. The truckload to food bank matching decisions are currently made by an employee of our partner organization. In addition to this being a very time-consuming task, as perhaps expected from human-based matching decisions, the allocations are often skewed: a small percentage of the possible recipients receives the majority of donations. Our goal in this partnership is to completely automate Food Drop. In doing so, we need a matching algorithm for making real-time decisions that strikes a balance between ensuring fairness for the food banks that receive the food and optimizing efficiency for the truck drivers. In this paper, we describe the theoretical guarantees and experiments that dictated our choice of algorithm in the platform we built and deployed for our partner organization. Our work also makes contributions to the literature on load balancing and balls-into-bins games, that might be of independent interest. Specifically, we study the allocation of $m$ weighted balls into $n$ weighted bins, where each ball has two non-uniformly sampled random bin choices, and prove upper bounds, that hold with high probability, on the maximum load of any bin.

cs.GT↗

arXiv2024

Getting More by Knowing Less: Bayesian Incentive Compatible Mechanisms for Fair Division

We study fair resource allocation with strategic agents. It is well-known that, across multiple fundamental problems in this domain, truthfulness and fairness are incompatible. For example, when allocating indivisible goods, no truthful and deterministic mechanism can guarantee envy-freeness up to one item (EF1), even for two agents with additive valuations. Or, in cake-cutting, no truthful and deterministic mechanism always outputs a proportional allocation, even for two agents with piecewise constant valuations. Our work stems from the observation that, in the context of fair division, truthfulness is used as a synonym for Dominant Strategy Incentive Compatibility (DSIC), requiring that an agent prefers reporting the truth, no matter what other agents report. In this paper, we instead focus on Bayesian Incentive Compatible (BIC) mechanisms, requiring that agents are better off reporting the truth in expectation over other agents' reports. We prove that, when agents know a bit less about each other, a lot more is possible: BIC mechanisms can guarantee fairness notions that are unattainable by DSIC mechanisms in both the fundamental problems of allocation of indivisible goods and cake-cutting. We prove that this is the case even for an arbitrary number of agents, as long as the agents' priors about each others' types satisfy a neutrality condition. Notably, for the case of indivisible goods, we significantly strengthen the state-of-the-art negative result for efficient DSIC mechanisms, while also highlighting the limitations of BIC mechanisms, by showing that a very general class of welfare objectives is incompatible with Bayesian Incentive Compatibility. Combined these results give a near-complete picture of the power and limitations of BIC and DSIC mechanisms for the problem of allocating indivisible goods.

cs.GT↗

arXiv2024

Fair and Efficient Allocations Without Obvious Manipulations

We consider the fundamental problem of allocating a set of indivisible goods among strategic agents with additive valuation functions. It is well known that, in the absence of monetary transfers, Pareto efficient and truthful rules are dictatorial, while there is no deterministic truthful mechanism that allocates all items and achieves envy-freeness up to one item (EF1), even for the case of two agents. In this paper, we investigate the interplay of fairness and efficiency under a relaxation of truthfulness called non-obvious manipulability (NOM), recently proposed by Troyan and Morrill. We show that this relaxation allows us to bypass the aforementioned negative results in a very strong sense. Specifically, we prove that there are deterministic and EF1 algorithms that are not obviously manipulable, and the algorithm that maximizes utilitarian social welfare (the sum of agents' utilities), which is Pareto efficient but not dictatorial, is not obviously manipulable for $n \geq 3$ agents (but obviously manipulable for $n=2$ agents). At the same time, maximizing the egalitarian social welfare (the minimum of agents' utilities) or the Nash social welfare (the product of agents' utilities) is obviously manipulable for any number of agents and items. Our main result is an approximation preserving black-box reduction from the problem of designing EF1 and NOM mechanisms to the problem of designing EF1 algorithms. En route, we prove an interesting structural result about EF1 allocations, as well as new "best-of-both-worlds" results (for the problem without incentives), that might be of independent interest.

cs.GT↗

arXiv2024

On the Fairness of Normalized p-Means for Allocating Goods and Chores

Allocating items in a fair and economically efficient manner is a central problem in fair division. We study this problem for agents with additive preferences, when items are all goods or all chores, divisible or indivisible. The celebrated notion of Nash welfare is known to produce fair and efficient allocations for both divisible and indivisible goods; there is no known analogue for dividing chores. The Nash welfare objective belongs to a large, parameterized family of objectives called the p-mean welfare functions, which includes other notable members, like social welfare and egalitarian welfare. However, among the members of this family, only the Nash welfare produces fair allocations for goods. Incidentally, Nash welfare is also the only member that satisfies the axiom of scale invariance, which is crucially associated with its fairness properties. We define the class of "normalized p-mean" objectives, which imparts the missing key axiom of scale invariance to the p-mean family. Our results show that optimizing the normalized p-mean objectives produces fair and efficient allocations when the items are goods or chores, divisible or indivisible. For instance, the normalized p-means gives us an infinite class of objectives that produce (i) proportional and Pareto efficient allocations for divisible goods, (ii) approximately proportional and Pareto efficient allocations for divisible chores, (iii) EF1 and Pareto efficient allocations for indivisible goods for two agents, and (iv) EF1 and Pareto efficient allocations for indivisible chores for two agents.

cs.GT↗

arXiv2023

Refined Mechanism Design for Approximately Structured Priors via Active Regression

We consider the problem of a revenue-maximizing seller with a large number of items $m$ for sale to $n$ strategic bidders, whose valuations are drawn independently from high-dimensional, unknown prior distributions. It is well-known that optimal and even approximately-optimal mechanisms for this setting are notoriously difficult to characterize or compute, and, even when they can be found, are often rife with various counter-intuitive properties. In this paper, following a model introduced recently by Cai and Daskalakis~\cite{cai2022recommender}, we consider the case that bidders' prior distributions can be well-approximated by a topic model. We design an active learning component, responsible for interacting with the bidders and outputting low-dimensional approximations of their types, and a mechanism design component, responsible for robustifying mechanisms for the low-dimensional model to work for the approximate types of the former component. On the active learning front, we cast our problem in the framework of Randomized Linear Algebra (RLA) for regression problems, allowing us to import several breakthrough results from that line of research, and adapt them to our setting. On the mechanism design front, we remove many restrictive assumptions of prior work on the type of access needed to the underlying distributions and the associated mechanisms. To the best of our knowledge, our work is the first to formulate connections between mechanism design, and RLA for active learning of regression problems, opening the door for further applications of randomized linear algebra primitives to mechanism design.

cs.GT↗

arXiv2023

On the Existence of Envy-Free Allocations Beyond Additive Valuations

We study the problem of fairly allocating $m$ indivisible items among $n$ agents. Envy-free allocations, in which each agent prefers her bundle to the bundle of every other agent, need not exist in the worst case. However, when agents have additive preferences and the value $v_{i,j}$ of agent $i$ for item $j$ is drawn independently from a distribution $D_i$, envy-free allocations exist with high probability when $m \in Ω( n \log n / \log \log n )$. In this paper, we study the existence of envy-free allocations under stochastic valuations far beyond the additive setting. We introduce a new stochastic model in which each agent's valuation is sampled by first fixing a worst-case function, and then drawing a uniformly random renaming of the items, independently for each agent. This strictly generalizes known settings; for example, $v_{i,j} \sim D_i$ may be seen as picking a random (instead of a worst-case) additive function before renaming. We prove that random renaming is sufficient to ensure that envy-free allocations exist with high probability in very general settings. When valuations are non-negative and ``order-consistent,'' a valuation class that generalizes additive, budget-additive, unit-demand, and single-minded agents, SD-envy-free allocations (a stronger notion of fairness than envy-freeness) exist for $m \in ω(n^2)$ when $n$ divides $m$, and SD-EFX allocations exist for all $m \in ω(n^2)$. The dependence on $n$ is tight, that is, for $m \in O(n^2)$ envy-free allocations don't exist with constant probability. For the case of arbitrary valuations (allowing non-monotone, negative, or mixed-manna valuations) and $n=2$ agents, we prove envy-free allocations exist with probability $1 - Θ(1/m)$ (and this is tight).

cs.GT↗

arXiv2023

Fair Chore Division under Binary Supermodular Costs

We study the problem of dividing indivisible chores among agents whose costs (for the chores) are supermodular set functions with binary marginals. Such functions capture complementarity among chores, i.e., they constitute an expressive class wherein the marginal disutility of each chore is either one or zero, and the marginals increase with respect to supersets. In this setting, we study the broad landscape of finding fair and efficient chore allocations. In particular, we establish the existence of $(i)$ EF1 and Pareto efficient chore allocations, $(ii)$ MMS-fair and Pareto efficient allocations, and $(iii)$ Lorenz dominating chore allocations. Furthermore, we develop polynomial-time algorithms--in the value oracle model--for computing the chore allocations for each of these fairness and efficiency criteria. Complementing these existential and algorithmic results, we show that in this chore division setting, the aforementioned fairness notions, namely EF1, MMS, and Lorenz domination are incomparable: an allocation that satisfies any one of these notions does not necessarily satisfy the others. Additionally, we study EFX chore division. In contrast to the above-mentioned positive results, we show that, for binary supermodular costs, Pareto efficient allocations that are even approximately EFX do not exist, for any arbitrarily small approximation constant. Focusing on EFX fairness alone, when the cost functions are identical we present an algorithm (Add-and-Fix) that computes an EFX allocation. For binary marginals, we show that Add-and-Fix runs in polynomial time.

cs.GT↗

arXiv2023

Decision-Focused Evaluation: Analyzing Performance of Deployed Restless Multi-Arm Bandits

Restless multi-arm bandits (RMABs) is a popular decision-theoretic framework that has been used to model real-world sequential decision making problems in public health, wildlife conservation, communication systems, and beyond. Deployed RMAB systems typically operate in two stages: the first predicts the unknown parameters defining the RMAB instance, and the second employs an optimization algorithm to solve the constructed RMAB instance. In this work we provide and analyze the results from a first-of-its-kind deployment of an RMAB system in public health domain, aimed at improving maternal and child health. Our analysis is focused towards understanding the relationship between prediction accuracy and overall performance of deployed RMAB systems. This is crucial for determining the value of investing in improving predictive accuracy towards improving the final system performance, and is useful for diagnosing, monitoring deployed RMAB systems. Using real-world data from our deployed RMAB system, we demonstrate that an improvement in overall prediction accuracy may even be accompanied by a degradation in the performance of RMAB system -- a broad investment of resources to improve overall prediction accuracy may not yield expected results. Following this, we develop decision-focused evaluation metrics to evaluate the predictive component and show that it is better at explaining (both empirically and theoretically) the overall performance of a deployed RMAB system.

cs.AI↗

arXiv2022

Constraining Brans-Dicke Parameter Using Gravitational Radiation

This article presents a theoretical limit on the Brans-Dicke (BD) parameter using a neutron star (NS) revolving around a galactic central engine.The assumption that the orbital radius is large enough allows to ignore strong curvature due to the central supermassive black hole (SMBH) and treat the problem with Newtonian dynamics. First, gravitational radiation polarizations are calculated in the BD theory. The scalar polarization, dominated by the dipole radiation, is then used to put a constraint on the BD parameter.The calculations are relatively simple and this makes it accessible to sophomore students.

gr-qc↗