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Jugal Garg

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44

AAAI Conference 2026 Conference Paper

Designing Truthful Mechanisms for Asymptotic Fair Division

  • Jugal Garg
  • Vishnu V. Narayan
  • Yuang Eric Shen

We study the problem of fairly allocating a set of m goods among n agents in the asymptotic setting, where each item's value for each agent is drawn from an underlying joint distribution. Prior works have shown that if this distribution is well-behaved, then an envy-free allocation exists with high probability when m=Ω(n log n). Under the stronger assumption that item values are independently and identically distributed (i.i.d.) across agents, it is known that this requirement improves to m=Ω(n log n / log log n), which is tight. However, these results rely on non-strategyproof mechanisms, such as maximum-welfare allocation or the round-robin algorithm, limiting their applicability in settings with strategic agents. In this work, we extend the theory to a broader, more realistic class of joint value distributions, allowing for correlations among agents, atomicity, and unequal probabilities of having the highest value for an item. We show that envy-free allocations continue to exist with a high probability when m=Ω(n log n). More importantly, we give a new randomized mechanism that is truthful in expectation, efficiently implementable in polynomial time, and outputs envy-free allocations with high probability, answering an open question from the literature. We further extend our mechanism to settings with asymptotic weighted fair division and multiple agent types and good types, proving new results in each case.

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AAAI Conference 2026 Conference Paper

Existence of 2-EFX Allocations of Chores

  • Jugal Garg
  • Aniket Murhekar

We study the fair division of indivisible chores among agents with additive disutility functions. We investigate the existence of allocations satisfying the popular fairness notion of envy-freeness up to any chore (EFX), and its multiplicative approximations. The existence of 4-EFX allocations was recently established by Garg, Murhekar, and Qin (2025). We improve this guarantee by proving the existence of 2-EFX allocations for all instances with additive disutilities. This approximation was previously known only for restricted instances such as bivalued disutilities (Lin, Wu, and Zhou (2025)) or three agents (Afshinmehr, Ansaripour, Danaei, and Mehlhorn (2024)). We obtain our result by providing a general framework for achieving approximate-EFX allocations. The approach begins with a suitable initial allocation and performs a sequence of local swaps between the bundles of envious and envied agents. For our main result, we begin with an initial allocation that satisfies envy-freeness up to one chore (EF1) and Pareto-optimality (PO); the existence of such an allocation was recently established in a major breakthrough by Mahara (2025). We further demonstrate the strength and generality of our framework by giving simple and unified proofs of existing results, namely (i) 2-EFX for bivalued instances, (ii) 2-EFX for three agents, (iii) EFX when the number of chores is at most twice the number of agents, and (iv) 4-EFX for all instances. We expect this framework to have broader applications in approximate-EFX due to its simplicity and generality.

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STOC Conference 2025 Conference Paper

Constant-Factor EFX Exists for Chores

  • Jugal Garg
  • Aniket Murhekar
  • John Qin

We study the problem of fair allocation of chores among 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 in fair division. The current best guarantee is the existence of O ( n 2 )-EFX allocations, where n denotes the number of agents, obtained through a sophisticated algorithm. In this paper, we show the existence of 4-EFX allocations, providing the first constant-factor approximation of EFX. We further 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, thereby improving upon 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-EF2 and PO. Our results are obtained via a novel economic framework called earning restricted (ER) competitive equilibrium for fractional allocations, which imposes limits on the earnings of agents from each chore. We show the existence of ER equilibria by carefully formulating a linear complementarity problem (LCP) that captures all ER equilibria, and then prove that the classic complementary pivot algorithm applied to this LCP terminates at an ER equilibrium. By carefully setting earning limits and leveraging the properties of ER equilibria, we design algorithms that involve rounding the fractional solutions and then performing swaps and merges of bundles to meet the desired fairness and efficiency criteria. We expect that the concept of ER equilibrium will play a crucial role in deriving further results on related problems.

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AAAI Conference 2025 Conference Paper

Improved Maximin Share Approximations for Chores by Bin Packing

  • Jugal Garg
  • Xin Huang
  • Erel Segal-Halevi

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 exist. We show the existence of · 1-out-of-9n/11 MMS allocations, which improves the state-of-the-art factor of 1-out-of-3n/4. · MMS allocations for factored instances, which resolves an open question posed by Ebadian et al. (2021). · 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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IJCAI Conference 2025 Conference Paper

Improved MMS Approximations for Few Agent Types

  • Parnian Shahkar
  • Jugal Garg

We study fair division of indivisible goods under the maximin share (MMS) fairness criterion in settings where agents are grouped into a small number of types, with agents within each type having identical valuations. For the special case of a single type, an exact MMS allocation is always guaranteed to exist. However, for two or more distinct agent types, exact MMS allocations do not always exist, shifting the focus to establishing the existence of approximate-MMS allocations. A series of works over the last decade has resulted in the best-known approximation guarantee of 3/4 + 3/3836. In this paper, we improve the approximation guarantees for settings where agents are grouped into two or three types, a scenario that arises in many practical settings. Specifically, we present novel algorithms that guarantee a 4/5-MMS allocation for two agent types and a 16/21-MMS allocation for three agent types. Our approach leverages the MMS partition of the majority type and adapts it to provide improved fairness guarantees for all types.

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AAMAS Conference 2025 Conference Paper

Matching Markets with Chores

  • Jugal Garg
  • Thorben Tröbst
  • Vijay V. Vazirani

The fair division of chores, as well as mixed manna (goods and chores), has received substantial recent attention in the fair division literature; however, ours is the first paper to extend this research to matching markets. Indeed, our contention is that matching markets are a natural setting for this purpose, since the manna that fit into the limited number of hours available in a day can be viewed as one unit of allocation. We extend several well-known results that hold for goods to the settings of chores and mixed manna. In addition, we show that the natural notion of an earnings-based equilibrium, which is more natural in the case of all chores, is equivalent to the pricing-based equilibrium given by Hylland and Zeckhauser for the case of goods.

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AAAI Conference 2025 Conference Paper

Proportionally Fair Makespan Approximation

  • Michal Feldman
  • Jugal Garg
  • Vishnu V. Narayan
  • Tomasz Ponitka

We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an Ω(log m / log log m)-approximation to the optimal makespan, where m is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a 3/2-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations.

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JAIR Journal 2024 Journal Article

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

  • Jugal Garg
  • Aniket Murhekar

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 pseudopolynomial 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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JAAMAS Journal 2024 Journal Article

One-sided matching markets with endowments: equilibria and algorithms

  • Jugal Garg
  • Thorben Tröbst
  • Vijay Vazirani

Abstract The Arrow–Debreu extension of the classic Hylland–Zeckhauser scheme (Hylland and Zeckhauser in J Polit Econ 87(2): 293–314, 1979) for a one-sided matching market—called ADHZ in this paper—has natural applications but has instances which do not admit equilibria. By introducing approximation, we define the \(\epsilon\) -approximate ADHZ model and give the following results. 1. Existence of equilibrium under linear utility functions. We prove that the equilibrium allocation satisfies Pareto optimality, approximate envy-freeness, and approximate weak core stability. 2. A combinatorial polynomial time algorithm for an \(\epsilon\) -approximate ADHZ equilibrium for the case of dichotomous, and more generally bi-valued, utilities. 3. An instance of ADHZ, with dichotomous utilities and a strongly connected demand graph, which does not admit an equilibrium. 4. A rational convex program for HZ under dichotomous utilities; a combinatorial polynomial time algorithm for this case was given in Vazirani and Yannakakis (in: Innovations in theoretical computer science, pp 59–15919, 2021). The \(\epsilon\) -approximate ADHZ model fills a void in the space of general mechanisms for one-sided matching markets; see details in the paper.

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IJCAI Conference 2024 Conference Paper

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

  • Jugal Garg
  • Aniket Murhekar
  • John Qin

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 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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AAMAS Conference 2023 Conference Paper

A Nash-Bargaining-Based Mechanism for One-Sided Matching Markets and Dichotomous Utilities

  • Jugal Garg
  • Thorben Tröbst
  • Vijay V. Vazirani

Mechanisms based on maximizing Nash Social Welfare (NSW) have proven to be fair and efficient for a wide variety of fair division problems. We study the fractional allocations maximizing NSW, i. e. , a Nash-bargaining-based mechanism, for one-sided matching markets with endowments, under dichotomous utilities, and show that they are the solutions of a rational convex program (RCP). Moreover, we provide a simple combinatorial polynomial time algorithm to maximize NSW by identifying the Nash bargaining points with the equilibrium of a novel type of market, the variable-budget market model. Lastly, we show that maximizing NSW is strategyproof under the assumption that the agents’ disagreement utilities are public knowledge.

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STOC Conference 2023 Conference Paper

Approximating Nash Social Welfare by Matching and Local Search

  • Jugal Garg
  • Edin Husic
  • Wenzheng Li
  • László A. Végh
  • Jan Vondrák

For any >0, we give a simple, deterministic (4+)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. The previous best approximation factor was 380 via a randomized algorithm. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents’ valuations, and give an (ω + 2 + ) -approximation if the ratio between the largest weight and the average weight is at most ω. We also show that the 12-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time which is both 12-EFX and a (8+)-approximation to the symmetric NSW problem under submodular valuations. The previous best approximation factor under 12-EFX was linear in the number of agents.

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JAIR Journal 2023 Journal Article

Competitive Equilibria with a Constant Number of Chores

  • Jugal Garg
  • Peter McGlaughlin
  • Martin Hoefer
  • Marco Schmalhofer

We study markets with mixed manna, where m divisible goods and chores shall be divided among n agents to obtain a competitive equilibrium. Equilibrium allocations are known to satisfy many fairness and efficiency conditions. While a lot of recent work in fair division is restricted to linear utilities and chores, we focus on a substantial generalization to separable piecewise-linear and concave (SPLC) utilities and mixed manna. We first derive polynomial-time algorithms for markets with a constant number of items or a constant number of agents. Our main result is a polynomial-time algorithm for instances with a constant number of chores (as well as any number of goods and agents) under the condition that chores dominate the utility of the agents. Interestingly, this stands in contrast to the case when the goods dominate the agents utility in equilibrium, where the problem is known to be PPAD-hard even without chores.

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IJCAI Conference 2023 Conference Paper

Fair and Efficient Allocation of Indivisible Chores with Surplus

  • Hannaneh Akrami
  • Bhaskar Ray Chaudhury
  • Jugal Garg
  • Kurt Mehlhorn
  • Ruta Mehta

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 in the chores setting 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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IJCAI Conference 2023 Conference Paper

New Algorithms for the Fair and Efficient Allocation of Indivisible Chores

  • Jugal Garg
  • Aniket Murhekar
  • John Qin

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 the 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 disutility functions. 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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IJCAI Conference 2023 Conference Paper

New Fairness Concepts for Allocating Indivisible Items

  • Ioannis Caragiannis
  • Jugal Garg
  • Nidhi Rathi
  • Eklavya Sharma
  • Giovanna Varricchio

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed till now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics. We propose two alternative fairness concepts—called epistemic EFX (EEFX) and minimum EFX value fairness (MXS)---inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts are excellent alternatives to EFX and MMS.

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IJCAI Conference 2023 Conference Paper

Simplification and Improvement of MMS Approximation

  • Hannaneh Akrami
  • Jugal Garg
  • Eklavya Sharma
  • Setareh Taki

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 (3/4 + 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 (3/4 + min(1/36, 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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SODA Conference 2022 Conference Paper

Approximating Equilibrium under Constrained Piecewise Linear Concave Utilities with Applications to Matching Markets

  • Jugal Garg
  • Yixin Tao
  • László A. Végh

We study the equilibrium computation problem in the Fisher market model with constrained piecewise linear concave (PLC) utilities. This general class captures many well-studied special cases, including markets with PLC utilities, markets with satiation, and matching markets. For the special case of PLC utilities, although the problem is PPAD-hard, Devanur and Kannan (FOCS 2008) gave a polynomial-time algorithm when the number of goods is constant. Our main result is a fixed parameter approximation scheme for computing an approximate equilibrium, where the parameters are the number of agents and the approximation accuracy. This provides an answer to an open question by Devanur and Kannan for PLC utilities, and gives a simpler and faster algorithm for matching markets as the one by Alaei, Jalaly and Tardos (EC 2017). The main technical idea is to work with the stronger concept of thrifty equilibria, and approximating the input utility functions by ‘robust’ utilities that have favorable marginal properties. With some restrictions, the results also extend to the Arrow–Debreu exchange market model.

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AAAI Conference 2022 Conference Paper

Fair and Efficient Allocations of Chores under Bivalued Preferences

  • Jugal Garg
  • Aniket Murhekar
  • John Qin

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 Paretooptimality (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 nontrivial 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 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 polynomialtime.

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JAIR Journal 2022 Journal Article

Fair Division of Indivisible Goods for a Class of Concave Valuations

  • Bhaskar Ray Chaudhury
  • Yun Kuen Cheung
  • Jugal Garg
  • Naveen Garg
  • Martin Hoefer
  • Kurt Mehlhorn

We study the fair and efficient allocation of a set of indivisible goods among agents, where each good has several copies, and each agent has an additively separable concave valuation function with a threshold. These valuations capture the property of diminishing marginal returns, and they are more general than the well-studied case of additive valuations. We present a polynomial-time algorithm that approximates the optimal Nash social welfare (NSW) up to a factor of e1/e ≈ 1.445. This matches with the state-of-the-art approximation factor for additive valuations. The computed allocation also satisfies the popular fairness guarantee of envy-freeness up to one good (EF1) up to a factor of 2 + ε. For instances without thresholds, it is also approximately Pareto-optimal. For instances satisfying a large market property, we show an improved approximation factor. Lastly, we show that the upper bounds on the optimal NSW introduced in Cole and Gkatzelis (2018) and Barman et al. (2018) have the same value.

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AAMAS Conference 2022 Conference Paper

One-Sided Matching Markets with Endowments: Equilibria and Algorithms

  • Jugal Garg
  • Thorben Tröbst
  • Vijay V. Vazirani

The Arrow-Debreu extension of the classic Hylland-Zeckhauser scheme [27] for a one-sided matching market – called ADHZ in this paper – has natural applications but has instances which do not admit equilibria. By introducing approximation, we define the 𝜖-approximate ADHZ model, and we give the following results. (1) Existence of equilibrium under linear utility functions. We prove that the equilibrium satisfies Pareto optimality, approximate envy-freeness, and approximate weak core stability. (2) A combinatorial polynomial time algorithm for an 𝜖- approximate ADHZ equilibrium for the case of dichotomous, and more generally bi-valued, utilities. (3) An instance of ADHZ, with dichotomous utilities and a strongly connected demand graph, which does not admit an equilibrium. (4) A rational convex program for HZ under dichotomous utilities; a combinatorial polynomial time algorithm for this case was given in [35]. The 𝜖-approximate ADHZ model fills a void in the space of general mechanisms for one-sided matching markets; see details in the paper.

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STOC Conference 2021 Conference Paper

Approximating Nash social welfare under rado valuations

  • Jugal Garg
  • Edin Husic
  • László A. Végh

We consider the problem of approximating maximum Nash social welfare (NSW) while allocating a set of indivisible items to n agents. The NSW is a popular objective that provides a balanced tradeoff between the often conflicting requirements of fairness and efficiency, defined as the weighted geometric mean of the agents’ valuations. For the symmetric additive case of the problem, where agents have the same weight with additive valuations, the first constant-factor approximation algorithm was obtained in 2015. Subsequent work has obtained constant-factor approximation algorithms for the symmetric case under mild generalizations of additive, and O ( n )-approximation algorithms for subadditive valuations and for the asymmetric case. In this paper, we make significant progress towards both symmetric and asymmetric NSW problems. We present the first constant-factor approximation algorithm for the symmetric case under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems, including as special cases assignment (OXS) valuations and weighted matroid rank functions. Furthermore, our approach also gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant.

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SODA Conference 2021 Conference Paper

Competitive Allocation of a Mixed Manna

  • Bhaskar Ray Chaudhury
  • Jugal Garg
  • Peter McGlaughlin
  • Ruta Mehta

We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads that everyone dislikes, as well as items that some like and others dislike. The seminal work of Bogomolnaia et al. [14] argue why allocating a mixed manna is genuinely more complicated than a good or a bad manna, and why competitive equilibrium is the best mechanism. They also provide the existence of equilibrium and establish its peculiar properties (e. g. , non-convex and disconnected set of equilibria even under linear utilities), but leave the problem of computing an equilibrium open. Our main result is a simplex-like algorithm based on Lemke's scheme for computing a competitive allocation of a mixed manna under SPLC utilities, a strict generalization of linear. Experimental results on randomly generated instances suggest that our algorithm will be fast in practice. The problem is known to be PPAD-hard for the case of good manna [24], and we also show a similar result for the case of bad manna. Given these PPAD-hardness results, designing such an algorithm is the only non-enumerative option known. Our algorithm also yields several new structural properties as simple corollaries. We obtain a (constructive) proof of existence for a far more general setting, membership of the problem in PPAD, rational-valued solution, and odd number of solutions property. The last property also settles the conjecture of [14] in the affirmative.

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AAAI Conference 2021 Conference Paper

Fair and Efficient Allocations under Subadditive Valuations

  • Bhaskar Ray Chaudhury
  • Jugal Garg
  • Ruta Mehta

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely 1 2 -EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents’ valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an O(n) approximation to the Nash welfare. Our result also improves the current best-known approximation of O(n log n) and O(m) to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an O(n) approximation to a family of welfare measures, p-mean of valuations for p ∈ (−∞, 1], thereby also matching asymptotically the current best approximation ratio for special cases like p = −∞ while also retaining the remarkable fairness properties.

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AAAI Conference 2021 Conference Paper

On Fair and Efficient Allocations of Indivisible Goods

  • Aniket Murhekar
  • Jugal Garg

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 pseudopolynomial time algorithm to compute an EF1+PO allocation, and a non-constructive proof of existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO). 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. We show that for such instances an EF1+fPO allocation can be computed in polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in polynomial time. Next, we consider instances where the number of agents is constant, and show that an EF1+PO (also 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. Further, we show that the problem of computing an EF1+PO allocation polynomial-time reduces to a problem in the complexity class PLS. We also design a polynomial-time algorithm that computes Nash welfare maximizing allocations when there are constantly many agents with constant many different values for the goods.

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SODA Conference 2020 Conference Paper

Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings

  • Jugal Garg
  • Pooja Kulkarni
  • Rucha Kulkarni

We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents' valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with factor independent of m is known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations. In this paper, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations respectively. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high valued items are done separately by un-matching certain items and re-matching them, by processes that are different in both algorithms. We show that these approaches achieve approximation factors of O ( n ) and O ( n log n ) for additive and submodular case respectively, which is independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1). Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of, hence resolving it completely.

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JAIR Journal 2020 Journal Article

Improving Nash Social Welfare Approximations

  • Peter McGlaughlin
  • Jugal Garg

We consider the problem of fairly allocating a set of indivisible goods among n agents. Various fairness notions have been proposed within the rapidly growing field of fair division, but the Nash social welfare (NSW) serves as a focal point. In part, this follows from the ‘unreasonable’ fairness guarantees provided, in the sense that a max NSW allocation meets multiple other fairness metrics simultaneously, all while satisfying a standard economic concept of efficiency, Pareto optimality. However, existing approximation algorithms fail to satisfy all of the remarkable fairness guarantees offered by a max NSW allocation, instead targeting only the specific NSW objective. We address this issue by presenting a 2 max NSW, Prop-1, 1/(2n) MMS, and Pareto optimal allocation in strongly polynomial time. Our techniques are based on a market interpretation of a fractional max NSW allocation. We present novel definitions of fairness concepts in terms of market prices, and design a new scheme to round a market equilibrium into an integral allocation in a way that provides most of the fairness properties of an integral max NSW allocation.

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IJCAI Conference 2019 Conference Paper

Improving Nash Social Welfare Approximations

  • Jugal Garg
  • Peter McGlaughlin

We consider the problem of fairly allocating a set of indivisible goods among n agents. Various fairness notions have been proposed within the rapidly growing field of fair division, but the Nash social welfare (NSW) serves as a focal point. In part, this follows from the 'unreasonable' fairness guarantees provided, in the sense that a max NSW allocation meets multiple other fairness metrics simultaneously, all while satisfying a standard economic concept of efficiency, Pareto optimality. However, existing approximation algorithms fail to satisfy all of the remarkable fairness guarantees offered by a max NSW allocation, instead targeting only the specific NSW objective. We address this issue by presenting a 2 max NSW, Prop-1, 1/(2n) MMS, and Pareto optimal allocation in strongly polynomial time. Our techniques are based on a market interpretation of a fractional max NSW allocation. We present novel definitions of fairness concepts in terms of market prices, and design a new scheme to round a market equilibrium into an integral allocation that provides most of the fairness properties of an integral max NSW allocation.

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SODA Conference 2018 Conference Paper

Approximating the Nash Social Welfare with Budget-Additive Valuations

  • Jugal Garg
  • Martin Hoefer 0001
  • Kurt Mehlhorn

We present the first constant-factor approximation algorithm for maximizing the Nash social welfare when allocating indivisible items to agents with budget-additive valuation functions. Budget-additive valuations represent an important class of submodular functions. They have attracted a lot of research interest in recent years due to many interesting applications. For every ε > 0, our algorithm obtains a (2. 404 + ε )-approximation in time polynomial in the input size and 1/ ε. Our algorithm relies on rounding an approximate equilibrium in a linear Fisher market where sellers have earning limits (upper bounds on the amount of money they want to earn) and buyers have utility limits (upper bounds on the amount of utility they want to achieve). In contrast to markets with either earning or utility limits, these markets have not been studied before. They turn out to have fundamentally different properties. Although the existence of equilibria is not guaranteed, we show that the market instances arising from the Nash social welfare problem always have an equilibrium. Further, we show that the set of equilibria is not convex, answering a question of [17]. We design an FPTAS to compute an approximate equilibrium, a result that may be of independent interest.

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SODA Conference 2016 Conference Paper

An Improved Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

  • Ran Duan
  • Jugal Garg
  • Kurt Mehlhorn

We present an improved combinatorial algorithm for the computation of equilibrium prices in the linear Arrow-Debreu model. For a market with n agents and integral utilities bounded by U, the algorithm runs in O ( n 7 log 3 ( nU )) time. This improves upon the previously best algorithm of Ye by a factor of. The algorithm refines the algorithm described by Duan and Mehlhorn and improves it by a factor of. The improvement comes from a better understanding of the iterative price adjustment process, the improved balanced flow computation for nondegenerate instances, and a novel perturbation technique for achieving nondegeneracy.

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AAAI Conference 2016 Conference Paper

Learning Market Parameters Using Aggregate Demand Queries

  • Xiaohui Bei
  • Wei Chen
  • Jugal Garg
  • Martin Hoefer
  • Xiaoming Sun

We study efficient algorithms for a natural learning problem in markets. There is one seller with m divisible goods and n buyers with unknown individual utility functions and budgets of money. The seller can repeatedly announce prices and observe aggregate demand bundles requested by the buyers. The goal of the seller is to learn the utility functions and budgets of the buyers. Our scenario falls into the classic domain of “revealed preference” analysis. Problems with revealed preference have recently started to attract increased interest in computer science due to their fundamental nature in understanding customer behavior in electronic markets. The goal of revealed preference analysis is to observe rational agent behavior, to explain it using a suitable model for the utility functions, and to predict future agent behavior. Our results are the first polynomial-time algorithms to learn utility and budget parameters via revealed preference queries in classic Fisher markets with multiple buyers. Our analysis concentrates on linear, CES, and Leontief markets, which are the most prominent classes studied in the literature. Some of our results extend to general Arrow-Debreu exchange markets.

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SODA Conference 2014 Conference Paper

On Computability of Equilibria in Markets with Production

  • Jugal Garg
  • Vijay V. Vazirani

Even though production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i. e. , the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. For markets with separable, piecewise-linear concave (SPLC) utilities and SPLC production, we obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria, and we further give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 [14]. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i. e. , without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm is practical. Also, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.

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