arXiv subjects

Jugal Garg

Publications and source records attributed to Jugal Garg.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the Theoretical Foundations of Data Exchange Economies

The immense success of ML systems relies heavily on large-scale, high-quality data. The high demand for data has led to many paradigms that involve selling, exchanging, and sharing data, motivating the study of economic processes with data as an asset. However, data differs from classical economic assets in terms of free duplication: there is no concept of limited supply since it can be replicated at zero marginal cost. This distinction introduces fundamental differences between economic processes involving data and those concerning other assets. We study a parallel to exchange (Arrow-Debreu) markets where data is the asset. Here, agents with datasets exchange data fairly and voluntarily, aiming for mutual benefit without monetary compensation. This framework is particularly relevant for non-profit organizations that seek to improve their ML models through data exchange, yet are restricted from selling their data for profit. We propose a general framework for data exchange, built on two core principles: (i) fairness, ensuring that each agent receives utility proportional to their contribution to others; contributions are quantifiable using standard credit-sharing functions like the Shapley value, and (ii) stability, ensuring that no coalition of agents can identify an exchange among themselves which they unanimously prefer to the current exchange. We show that fair and stable exchanges exist for all monotone continuous utility functions. Next, we investigate the computational complexity of finding approximate fair and stable exchanges. We present a local search algorithm for instances with monotone submodular utility functions, where each agent contributions are measured using the Shapley value. We prove that this problem lies in CLS under mild assumptions. Our framework opens up several intriguing theoretical directions for research in data economics.

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arXiv2024

Constant-Factor EFX Exists for Chores

We study the problem of fair allocation of chores to agents with additive preferences. In the discrete setting, envy-freeness up to any chore (EFX) has emerged as a compelling fairness criterion. However, establishing its (non-)existence or achieving a meaningful approximation remains a major open question. The current best guarantee is the existence of $O(n^2)$-EFX allocations for $n$ agents, obtained through a sophisticated algorithm (Zhou and Wu, 2022). In this paper, we show the existence of $4$-EFX allocations, providing the first constant-factor approximation of EFX. We also investigate the existence of allocations that are both fair and efficient, using Pareto optimality (PO) as our efficiency criterion. For the special case of bivalued instances, we establish the existence of allocations that are both $3$-EFX and PO, thus improving the current best factor of $O(n)$-EFX without any efficiency guarantees. For general additive instances, the existence of allocations that are $α$-EF$k$ and PO has remained open for any constant values of $α$ and $k$, where EF$k$ denotes envy-freeness up to $k$ chores. We provide the first positive result in this direction by showing the existence of allocations that are $2$-EF$2$ and PO. Our results are obtained via a novel economic framework called earning restricted (ER) competitive equilibrium for fractional allocations, which limits agents' earnings from each chore. We show the existence of ER equilibria by formulating it as an linear complementarity problem (LCP) and proving that the classic complementary pivot algorithm on the LCP terminates at an ER equilibrium. We design algorithms that carefully round fractional ER equilibria, and perform bundle swaps and merges to meet the desired fairness and efficiency criteria. We expect that the concept of ER equilibrium will be useful in deriving further results on related problems.

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arXiv2024

Improved Maximin Share Approximations for Chores by Bin Packing

We study fair division of indivisible chores among $n$ agents with additive cost functions using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist for more than two agents, the goal has been to improve its approximations and identify interesting special cases where MMS allocations exists. We show the existence of 1) 1-out-of-$\lfloor \frac{9}{11}n\rfloor$ MMS allocations, which improves the state-of-the-art factor of 1-out-of-$\lfloor \frac{3}{4}n\rfloor$. 2) MMS allocations for factored instances, which resolves an open question posed by Ebadian et al. (2021). 3) $15/13$-MMS allocations for personalized bivalued instances, improving the state-of-the-art factor of $13/11$. We achieve these results by leveraging the HFFD algorithm of Huang and Lu (2021). Our approach also provides polynomial-time algorithms for computing an MMS allocation for factored instances and a $15/13$-MMS allocation for personalized bivalued instances.

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arXiv2024

EF1 for Mixed Manna with Unequal Entitlements

We study fair division of indivisible mixed manna when agents have unequal entitlements, with weighted envy-freeness up to one item (WEF1) as our primary notion of fairness. We identify several shortcomings of existing techniques to achieve WEF1. Hence, we relax WEF1 to weighted envy-freeness up to 1 transfer (WEF1T), and give a polynomial-time algorithm for achieving it. We also generalize Fisher markets to the mixed manna setting, and use them to get a polynomial-time algorithm for two agents that outputs a WEF1 allocation.

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arXiv2024

Weighted EF1 and PO Allocations with Few Types of Agents or Chores

We investigate the existence of fair and efficient allocations of indivisible chores to asymmetric agents who have unequal entitlements or weights. We consider the fairness notion of weighted envy-freeness up to one chore (wEF1) and the efficiency notion of Pareto-optimality (PO). The existence of EF1 and PO allocations of chores to symmetric agents is a major open problem in discrete fair division, and positive results are known only for certain structured instances. In this paper, we study this problem for a more general setting of asymmetric agents and show that an allocation that is wEF1 and PO exists and can be computed in polynomial time for instances with: - Three types of agents, where agents with the same type have identical preferences but can have different weights. - Two types of chores, where the chores can be partitioned into two sets, each containing copies of the same chore. For symmetric agents, our results establish that EF1 and PO allocations exist for three types of agents and also generalize known results for three agents, two types of agents, and two types of chores. Our algorithms use a weighted picking sequence algorithm as a subroutine; we expect this idea and our analysis to be of independent interest.

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arXiv2024

Improving Approximation Guarantees for Maximin Share

We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her ($1$-out-of-$n$) MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of $1$-out-of-$d$ MMS allocations (for the smallest possible $d>n$). A series of works led to the state-of-the-art factor of $d=\lfloor3n/2\rfloor$ [Hosseini et al.'21]. We show that $1$-out-of-$4\lceil n/3\rceil$ MMS allocations always exist, thereby improving the state-of-the-art of ordinal approximation. In the multiplicative approximation, the goal is to show the existence of $α$-MMS allocations (for the largest possible $α< 1$), which guarantees each agent at least $α$ times her MMS value. We introduce a general framework of "approximate MMS with agent priority ranking". An allocation is said to be $T$-MMS, for a non-increasing sequence $T = (τ_1, \ldots, τ_n)$ of numbers, if the agent at rank $i$ in the order gets a bundle of value at least $τ_i$ times her MMS value. This framework captures both ordinal approximation and multiplicative approximation as special cases. We show the existence of $T$-MMS allocations where $τ_i \ge \max(\frac{3}{4} + \frac{1}{12n}, \frac{2n}{2n+i-1})$ for all $i$. Furthermore, we can get allocations that are $(\frac{3}{4} + \frac{1}{12n})$-MMS ex-post and $(0.8253 + \frac{1}{36n})$-MMS ex-ante. We also prove that our algorithm does not give better than $(0.8631 + \frac{1}{2n})$-MMS ex-ante.

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arXiv2024

Limited-Trust in Social Network Games

We consider agents in a social network competing to be selected as partners in collaborative, mutually beneficial activities. We study this through a model in which an agent i can initiate a limited number k_i>0 of games and selects the ideal partners from its one-hop neighborhood. On the flip side it can accept as many games offered from its neighbors. Each game signifies a productive joint economic activity, and players attempt to maximize their individual utilities. Unsurprisingly, more trustworthy agents are more desirable as partners. Trustworthiness is measured by the game theoretic concept of Limited-Trust, which quantifies the maximum cost an agent is willing to incur in order to improve the net utility of all agents. Agents learn about their neighbors' trustworthiness through interactions and their behaviors evolve in response. Empirical trials performed on realistic social networks show that when given the option, many agents become highly trustworthy; most or all become highly trustworthy when knowledge of their neighbors' trustworthiness is based on past interactions rather than known a priori. This trustworthiness is not the result of altruism, instead agents are intrinsically motivated to become trustworthy partners by competition. Two insights are presented: first, trustworthy behavior drives an increase in the utility of all agents, where maintaining a relatively modest level of trustworthiness may easily improve net utility by as much as 14.5%. If only one agent exhibits modest trust among self-centered ones, it can increase its average utility by up to 25% in certain cases! Second, and counter-intuitively, when partnership opportunities are abundant agents become less trustworthy.

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arXiv2023

Computing Pareto-Optimal and Almost Envy-Free Allocations of Indivisible Goods

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of $k$-ary instances where $k$ is a constant, i.e., each agent has at most $k$ different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.

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arXiv2023

New Algorithms for the Fair and Efficient Allocation of Indivisible Chores

We study the problem of fairly and efficiently allocating indivisible chores among agents with additive disutility functions. We consider the widely-used envy-based fairness properties of EF1 and EFX, in conjunction with the efficiency property of fractional Pareto-optimality (fPO). Existence (and computation) of an allocation that is simultaneously EF1/EFX and fPO are challenging open problems, and we make progress on both of them. We show existence of an allocation that is - EF1+fPO, when there are three agents, - EF1+fPO, when there are at most two disutility functions, - EFX+fPO, for three agents with bivalued disutilities. These results are constructive, based on strongly polynomial-time algorithms. We also investigate non-existence and show that an allocation that is EFX+fPO need not exist, even for two agents.

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arXiv2023

Breaking the $3/4$ Barrier for Approximate Maximin Share

We study the fundamental problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. Since MMS allocations need not exist when $n>2$, a series of works showed the existence of approximate MMS allocations with the current best factor of $\frac34 + O(\frac{1}{n})$. However, a simple example in [DFL82, BEF21, AGST23] showed the limitations of existing approaches and proved that they cannot improve this factor to $3/4 + Ω(1)$. In this paper, we bypass these barriers to show the existence of $(\frac{3}{4} + \frac{3}{3836})$-MMS allocations by developing new reduction rules and analysis techniques.

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arXiv2023

Simplification and Improvement of MMS Approximation

We consider the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of $(\frac{3}{4} + \frac{1}{12n})$. Most of these results are based on complicated analyses, especially those providing better than $2/3$ factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. For small $n$, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.

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arXiv2023

Fair and Efficient Allocation of Indivisible Chores with Surplus

We study fair division of indivisible chores among $n$ agents with additive disutility functions. Two well-studied fairness notions for indivisible items are envy-freeness up to one/any item (EF1/EFX) and the standard notion of economic efficiency is Pareto optimality (PO). There is a noticeable gap between the results known for both EF1 and EFX in the goods and chores settings. The case of chores turns out to be much more challenging. We reduce this gap by providing slightly relaxed versions of the known results on goods for the chores setting. Interestingly, our algorithms run in polynomial time, unlike their analogous versions in the goods setting. We introduce the concept of $k$ surplus which means that up to $k$ more chores are allocated to the agents and each of them is a copy of an original chore. We present a polynomial-time algorithm which gives EF1 and PO allocations with $(n-1)$ surplus. We relax the notion of EFX slightly and define tEFX which requires that the envy from agent $i$ to agent $j$ is removed upon the transfer of any chore from the $i$'s bundle to $j$'s bundle. We give a polynomial-time algorithm that in the chores case for $3$ agents returns an allocation which is either proportional or tEFX. Note that proportionality is a very strong criterion in the case of indivisible items, and hence both notions we guarantee are desirable.

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arXiv2022

EFX Allocations: Simplifications and Improvements

The existence of EFX allocations is a fundamental open problem in discrete fair division. Given a set of agents and indivisible goods, the goal is to determine the existence of an allocation where no agent envies another following the removal of any single good from the other agent's bundle. Since the general problem has been illusive, progress is made on two fronts: $(i)$ proving existence when the number of agents is small, $(ii)$ proving existence of relaxations of EFX. In this paper, we improve results on both fronts (and simplify in one of the cases). We prove the existence of EFX allocations with three agents, restricting only one agent to have an MMS-feasible valuation function (a strict generalization of nice-cancelable valuation functions introduced by Berger et al. which subsumes additive, budget-additive and unit demand valuation functions). The other agents may have any monotone valuation functions. Our proof technique is significantly simpler and shorter than the proof by Chaudhury et al. on existence of EFX allocations when there are three agents with additive valuation functions and therefore more accessible. Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX allocations and EFX allocations with few unallocated goods (charity). Chaudhury et al. showed the existence of $(1-ε)$-EFX allocation with $O((n/ε)^{\frac{4}{5}})$ charity by establishing a connection to a problem in extremal combinatorics. We improve their result and prove the existence of $(1-ε)$-EFX allocations with $\tilde{O}((n/ ε)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be used to prove improved upper-bounds on a problem in zero-sum combinatorics introduced by Alon and Krivelevich.

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arXiv2022

Competitive Equilibrium with Chores: Combinatorial Algorithm and Hardness

We study the computational complexity of finding a competitive equilibrium (CE) with chores when agents have linear preferences. CE is one of the most preferred mechanisms for allocating a set of items among agents. CE with equal incomes (CEEI), Fisher, and Arrow-Debreu (exchange) are the fundamental economic models to study allocation problems, where CEEI is a special case of Fisher and Fisher is a special case of exchange. When the items are goods (giving utility), the CE set is convex even in the exchange model, facilitating several combinatorial polynomial-time algorithms (starting with the seminal work of Devanur, Papadimitriou, Saberi and Vazirani) for all of these models. In sharp contrast, when the items are chores (giving disutility), the CE set is known to be non-convex and disconnected even in the CEEI model. Further, no combinatorial algorithms or hardness results are known for these models. In this paper, we give two main results for CE with chores: 1) A combinatorial algorithm to compute a $(1-\varepsilon)$-approximate CEEI in time $\tilde{\mathcal{O}}(n^4m^2 / \varepsilon^2)$, where $n$ is the number of agents and $m$ is the number of chores. 2) PPAD-hardness of finding a $(1-1/\mathit{poly}(n))$-approximate CE in the exchange model under a sufficient condition. To the best of our knowledge, these results show the first separation between the CEEI and exchange models when agents have linear preferences, assuming PPAD $\neq $ P. Finally, we show that our new insight implies a straightforward proof of the existence of an allocation that is both envy-free up to one chore (EF1) and Pareto optimal (PO) in the discrete setting when agents have factored bivalued preferences.

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arXiv2022

Auction Algorithms for Market Equilibrium with Weak Gross Substitute Demands

We consider the Arrow--Debreu exchange market model under the assumption that the agents' demands satisfy the weak gross substitutes (WGS) property. We present a simple auction algorithm that obtains an approximate market equilibrium for WGS demands assuming the availability of a price update oracle. We exhibit specific implementations of such an oracle for WGS demands with bounded price elasticities and for Gale demand systems. As an application of our result, we obtain an efficient algorithm to find an approximate spending-restricted market equilibrium for WGS demands, a model that has been recently introduced as a continuous relaxation of the Nash social welfare (NSW) problem. This leads to a polynomial-time constant factor approximation algorithm for the NSW problem with capped additive separable piecewise linear utility functions; only a pseudopolynomial approximation algorithm was known for this setting previously.

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arXiv2022

Tractable Fragments of the Maximum Nash Welfare Problem

We study the problem of maximizing Nash welfare (MNW) while allocating indivisible goods to asymmetric agents. The Nash welfare of an allocation is the weighted geometric mean of agents' utilities, and the allocation with maximum Nash welfare is known to satisfy several desirable fairness and efficiency properties. However, computing such an MNW allocation is NP-hard, even for two agents with identical, additive valuations. Hence, we aim to identify tractable classes that either admit a PTAS, an FPTAS, or an exact polynomial-time algorithm. To this end, we design a PTAS for finding an MNW allocation for the case of asymmetric agents with identical, additive valuations, thus generalizing a similar result for symmetric agents. Our techniques can also be adapted to give a PTAS for the problem of computing the optimal $p$-mean welfare. We also show that an MNW allocation can be computed exactly in polynomial time for identical agents with $k$-ary valuations when $k$ is a constant, where every agent has at most $k$ different values for the goods. Next, we consider the special case where every agent finds at most two goods valuable, and show that this class admits an efficient algorithm, even for general monotone valuations. In contrast, we note that when agents can value three or more goods, maximizing Nash welfare is NP-hard, even when agents are symmetric and have additive valuations, showing our algorithmic result is essentially tight. Finally, we show that for constantly many asymmetric agents with additive valuations, the MNW problem admits an FPTAS.

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arXiv2022

A Strongly Polynomial Algorithm for Linear Exchange Markets

We present a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities. Our algorithm is based on a variant of the weakly-polynomial Duan-Mehlhorn (DM) algorithm. We use the DM algorithm as a subroutine to identify revealed edges, i.e. pairs of agents and goods that must correspond to best bang-per-buck transactions in every equilibrium solution. Every time a new revealed edge is found, we use another subroutine that decides if there is an optimal solution using the current set of revealed edges, or if none exists, finds the solution that approximately minimizes the violation of the demand and supply constraints. This task can be reduced to solving a linear program (LP). Even though we are unable to solve this LP in strongly polynomial time, we show that it can be approximated by a simpler LP with two variables per inequality that is solvable in strongly polynomial time.

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arXiv2021

Fair and Efficient Allocations of Chores under Bivalued Preferences

We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that an EF1+PO allocation exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores. Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To the best of our knowledge, this is the first non-trivial class of indivisible chores to admit an EF1+PO allocation and an efficient algorithm for its computation. We also study the problem of computing an envy-free (EF) and PO allocation for the case of divisible chores. While the existence of an EF+PO allocation is known via competitive equilibrium with equal incomes, its efficient computation is open. Our second result shows that for bivalued instances, an EF+PO allocation can be computed in strongly polynomial-time.

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