arXiv subjects

Siddharth Barman

Publications and source records attributed to Siddharth Barman.

At least 19 recordsLinked to original sources

arXiv2026

Packing Linear Programs and Fractional Knapsack using Comparison Oracles

We study the problem of recovering the objective of a packing linear program when the algorithm accesses only comparison information about optimal solutions under varying constraint matrices. Motivated by optimization with comparison oracles (Cohen-Addad et al., STOC 2026) and preference feedback (Kaufmann et al., TMLR 2025), this strengthens inverse-optimization frameworks by replacing direct observations of optimal solutions with ordinal queries. We focus on the fractional knapsack problem, where the packing linear program (LP) has a single budget constraint specified by item prices, and the objective is determined by item values. This captures monopoly-pricing where a seller infers a buyer's unknown valuations for divisible items from comparison information. The algorithm queries an oracle with two price vectors, returning which optimal solution has the larger total packing or objective value. Such oracles abstract discrete-choice surveys of buyers choosing between differently priced alternatives. For fractional knapsack, we develop a polynomial-time algorithm recovering item values up to scale using $O(n \log(1/\delta)+B^2)$ comparison queries, where $n$ is the number of items, $B$ is the knapsack capacity, and $\delta$ is the value grid resolution. We complement this with an $\Omega(n \log(1/\delta))$ lower bound. A key insight is that in the comparison-oracle model, fractional knapsack is as general as packing LPs. Our algorithm solves the packing setting by treating a constraint matrix row as the price vector and zeroing the rest. The $\Omega(n \log(1/\delta))$ lower bound continues to hold for packing LPs, making our upper bound essentially best possible, up to a linear-factor gap. Finally, we extend our algorithm to profit-maximization, yielding a comparison-oracle analogue of the revealed-preference result of Amin et al. (AAAI 2015).

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arXiv2026

Equitable Colorings of Vertex-Weighted Graphs

We study a generalization of the classical Hajnal-Szemer\'edi theorem to vertex-weighted graphs. Given a graph with nonnegative vertex weights, a coloring is called $\alpha$-approximately equitable up to one vertex ($\alpha$-EQ1) if, for each color class, the total weight remaining after removing its maximum-weight vertex is at most $\alpha \geq 1$ times the weight of any other color class. For vertex-weighted graphs with maximum degree $\Delta$, we show that there exist instances for which no $k$-coloring is $\alpha$-EQ1 for any $k < \frac{3\Delta}{2}$ and $\alpha < \sqrt{2}$. In light of this impossibility, we relax these parameters and establish the following results for any vertex-weighted graph $G$ with maximum degree $\Delta$: (1) for any $\varepsilon \in (0,1)$ and all $k \geq (\frac{c}{\varepsilon^2}\ln{\frac{1}{\varepsilon}}) \Delta$, there exists a $(1 + \varepsilon)$-EQ1 $k$-coloring of $G$, where $c$ is a fixed constant; and (2) for all $k \ge \Delta + 1$, there exists a $2$-EQ1 $k$-coloring of $G$. Furthermore, such equitable colorings can be computed in polynomial time. En route to our results on equitability under vertex weights, we establish sufficient conditions for the existence of $k$-colorings that are equitable with respect to any given partition of the vertex set. Our coloring results correspond to fairness guarantees in a constrained fair division setting and lead to concentration inequalities for partly dependent random variables.

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arXiv2026

Existence and Computation of Fair Allocations under Constraints

We study fair division of divisible goods under generalized assignment constraints. Here, each good has an agent-specific value and size, and every agent has a budget constraint that limits the total size of the goods she can receive. Since it may not always be feasible to assign all goods to the agents while respecting the budget constraints, we use the construct of charity to accommodate the unassigned goods. In this constrained setting with charity, we obtain several new existential and computational results for feasible envy-freeness (FEF); this fairness notion requires that agents are envy-free, considering only budget-feasible subsets. First, we simplify and extend known existential results for FEF allocations. Then, we show that the space of FEF allocations has a non-convex structure. Next, using a fixed-point argument, we establish a novel guarantee that FEF can always be achieved with Pareto-optimality. Furthermore, we give an alternative proof of the fact that one cannot additionally obtain truthfulness in this context: There does not exist a mechanism that is simultaneously truthful, fair, and Pareto-optimal. On the positive side, we show that truthfulness is compatible with each of FEF and Pareto-optimality, individually.

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arXiv2026

Welfarist Formulations for Diverse Similarity Search

Nearest Neighbor Search (NNS) is a fundamental problem in data structures with wide-ranging applications, such as web search, recommendation systems, and, more recently, retrieval-augmented generations (RAG). In such recent applications, in addition to the relevance (similarity) of the returned neighbors, diversity among the neighbors is a central requirement. In this paper, we develop principled welfare-based formulations in NNS for realizing diversity across attributes. Our formulations are based on welfare functions -- from mathematical economics -- that satisfy central diversity (fairness) and relevance (economic efficiency) axioms. With a particular focus on Nash social welfare, we note that our welfare-based formulations provide objective functions that adaptively balance relevance and diversity in a query-dependent manner. Notably, such a balance was not present in the prior constraint-based approach, which forced a fixed level of diversity and optimized for relevance. In addition, our formulation provides a parametric way to control the trade-off between relevance and diversity, providing practitioners with flexibility to tailor search results to task-specific requirements. We develop efficient nearest neighbor algorithms with provable guarantees for the welfare-based objectives. Notably, our algorithm can be applied on top of any standard ANN method (i.e., use standard ANN method as a subroutine) to efficiently find neighbors that approximately maximize our welfare-based objectives. Experimental results demonstrate that our approach is practical and substantially improves diversity while maintaining high relevance of the retrieved neighbors.

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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.

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arXiv2025

Fair and Efficient Allocation of Indivisible Mixed Manna

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

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arXiv2025

Fair Division via Resource Augmentation

We introduce and formalize the notion of resource augmentation for maximin share (MMS) fairness for the allocation of indivisible goods. Given an instance with $n$ agents and $m$ goods, we ask how many copies of the goods should be added in order to guarantee that each agent receives at least their original MMS value, or a meaningful approximation thereof. For general monotone valuations, we establish a tight bound: an exact MMS allocation can be guaranteed using at most $\Theta(m/e)$ total copies, and this bound is tight even for XOS valuations. We further show that it is unavoidable to duplicate some goods $\Omega(\ln m / \ln \ln m)$ times, and provide matching upper bounds. For additive valuations, we show that at most $\min\{n-2,\lfloor\frac{m}{3}\rfloor(1+o(1))\}$ distinct copies suffice. This separates additive valuations from submodular valuations, for which we show that $n-1$ copies may be necessary. We also study approximate MMS guarantees for additive valuations and establish new tradeoffs between the number of copies needed and the approximation guaratee. In particular, we prove that $\lfloor{n/2}\rfloor$ copies suffice to guarantee a $6/7$-approximation to the original MMS, and $\lfloor{n/3}\rfloor$ copies suffice for a $4/5$-approximation. Both results improve upon the best-known approximation guarantees for additive valuations in the absence of copies. Finally, we relate MMS with copies to the relaxed notion of 1-out-of-$d$ MMS, showing that improvements in either framework translate directly to the other. In particular, we establish the first impossibility results for 1-out-of-$d$ MMS. Our results highlight the power and limits of resource augmentation for achieving MMS fairness.

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arXiv2025

Exact Maximin Share Fairness via Adjusted Supply

This work addresses fair allocation of indivisible items in settings wherein it is feasible to create copies of resources or dispose of tasks. We establish that exact maximin share (MMS) fairness can be achieved via limited duplication of goods even under monotone valuations. We also show that, when allocating chores under monotone costs, MMS fairness is always feasible with limited disposal of chores. Since monotone valuations do not admit any nontrivial approximation guarantees for MMS, our results highlight that such barriers can be circumvented by post facto adjustments in the supply of the items. We prove that, for division of $m$ goods among $n$ agents with monotone valuations, there always exists an assignment of subsets of goods to the agents such that they receive at least their maximin shares and no single good is allocated to more than $3 \log m$ agents. In addition, the sum of the sizes of the assigned subsets does not exceed $m$. For identically ordered valuations, we obtain an upper bound of $O(\sqrt{\log m})$ on the maximum assignment multiplicity across goods and an $m + \widetilde{O}\left(\frac{m}{\sqrt{n}} \right)$ bound for the total number of goods assigned. Further, for additive valuations, we prove that there always exists an MMS assignment in which no single good is allocated to more than $2$ agents and the total number of goods assigned is at most $2m$. For chores, we upper bound the number of chores that need to be discarded for ensuring MMS fairness. We prove that, under monotone costs, there exists an MMS assignment in which at most $\frac{m}{e}$ remain unassigned. For identically ordered costs, we establish that MMS fairness can be achieved while keeping at most $\widetilde{O} \left(\frac{m}{n^{1/4}} \right)$ chores unassigned. We also prove that the obtained bounds for monotone valuations and monotone costs are essentially tight.

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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.

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arXiv2024

Fair Division with Market Values

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and with respect to the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

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arXiv2024

Compatibility of Fairness and Nash Welfare under Subadditive Valuations

We establish a compatibility between fairness and efficiency, captured via Nash Social Welfare (NSW), under the broad class of subadditive valuations. We prove that, for subadditive valuations, there always exists a partial allocation that is envy-free up to the removal of any good (EFx) and has NSW at least half of the optimal; here, optimality is considered across all allocations, fair or otherwise. We also prove, for subadditive valuations, the universal existence of complete allocations that are envy-free up to one good (EF1) and also achieve a factor $1/2$ approximation to the optimal NSW. Our EF1 result resolves an open question posed by Garg, Husic, Li, V\'{e}gh, and Vondr\'{a}k (STOC 2023). In addition, we develop a polynomial-time algorithm which, given an arbitrary allocation $\widetilde{A}$ as input, returns an EF1 allocation with NSW at least $\frac{1}{e^{2/e}}\approx \frac{1}{2.08}$ times that of $\widetilde{A}$. Therefore, our results imply that the EF1 criterion can be attained simultaneously with a constant-factor approximation to optimal NSW in polynomial time (with demand queries), for subadditive valuations. The previously best-known approximation factor for optimal NSW, under EF1 and among $n$ agents, was $O(n)$ -- we improve this bound to $O(1)$. It is known that EF1 and exact Pareto efficiency (PO) are incompatible with subadditive valuations. Complementary to this negative result, the current work shows that we regain compatibility by just considering a factor $1/2$ approximation: EF1 can be achieved in conjunction with $\frac{1}{2}$-PO under subadditive valuations. As such, our results serve as a general tool that can be used as a black box to convert any efficient outcome into a fair one, with only a marginal decrease in efficiency.

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arXiv2024

Causal Contextual Bandits with Adaptive Context

We study a variant of causal contextual bandits where the context is chosen based on an initial intervention chosen by the learner. At the beginning of each round, the learner selects an initial action, depending on which a stochastic context is revealed by the environment. Following this, the learner then selects a final action and receives a reward. Given $T$ rounds of interactions with the environment, the objective of the learner is to learn a policy (of selecting the initial and the final action) with maximum expected reward. In this paper we study the specific situation where every action corresponds to intervening on a node in some known causal graph. We extend prior work from the deterministic context setting to obtain simple regret minimization guarantees. This is achieved through an instance-dependent causal parameter, $\lambda$, which characterizes our upper bound. Furthermore, we prove that our simple regret is essentially tight for a large class of instances. A key feature of our work is that we use convex optimization to address the bandit exploration problem. We also conduct experiments to validate our theoretical results, and release our code at our project GitHub repository: https://github.com/adaptiveContextualCausalBandits/aCCB.

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arXiv2024

Generalized Linear Bandits with Limited Adaptivity

We study the generalized linear contextual bandit problem within the constraints of limited adaptivity. In this paper, we present two algorithms, $\texttt{B-GLinCB}$ and $\texttt{RS-GLinCB}$, that address, respectively, two prevalent limited adaptivity settings. Given a budget $M$ on the number of policy updates, in the first setting, the algorithm needs to decide upfront $M$ rounds at which it will update its policy, while in the second setting it can adaptively perform $M$ policy updates during its course. For the first setting, we design an algorithm $\texttt{B-GLinCB}$, that incurs $\tilde{O}(\sqrt{T})$ regret when $M = \Omega( \log{\log T} )$ and the arm feature vectors are generated stochastically. For the second setting, we design an algorithm $\texttt{RS-GLinCB}$ that updates its policy $\tilde{O}(\log^2 T)$ times and achieves a regret of $\tilde{O}(\sqrt{T})$ even when the arm feature vectors are adversarially generated. Notably, in these bounds, we manage to eliminate the dependence on a key instance dependent parameter $\kappa$, that captures non-linearity of the underlying reward model. Our novel approach for removing this dependence for generalized linear contextual bandits might be of independent interest.

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arXiv2023

Parameterized Guarantees for Almost Envy-Free Allocations

We study fair allocation of indivisible goods among agents with additive valuations. We obtain novel approximation guarantees for three of the strongest fairness notions in discrete fair division, namely envy-free up to the removal of any positively-valued good (EFx), pairwise maximin shares (PMMS), and envy-free up to the transfer of any positively-valued good (tEFx). Our approximation guarantees are in terms of an instance-dependent parameter $\gamma \in (0,1]$ that upper bounds, for each indivisible good in the given instance, the multiplicative range of nonzero values for the good across the agents. First, we consider allocations wherein, between any pair of agents and up to the removal of any positively-valued good, the envy is multiplicatively bounded. Specifically, the current work develops a polynomial-time algorithm that computes a $\left( \frac{2\gamma}{\sqrt{5+4\gamma}-1}\right)$-approximately EFx allocation for any given fair division instance with range parameter $\gamma \in (0,1]$. For instances with $\gamma \geq 0.511$, the obtained approximation guarantee for EFx surpasses the previously best-known approximation bound of $(\phi-1) \approx 0.618$, here $\phi$ denotes the golden ratio. Furthermore, for $\gamma \in (0,1]$, we develop a polynomial-time algorithm for finding allocations wherein the PMMS requirement is satisfied, between every pair of agents, within a multiplicative factor of $\frac{5}{6} \gamma$. En route to this result, we obtain novel existential and computational guarantees for $\frac{5}{6}$-approximately PMMS allocations under restricted additive valuations. Finally, we develop an algorithm that efficiently computes a $2\gamma$-approximately tEFx allocation. Specifically, we obtain existence and efficient computation of exact tEFx allocations for all instances with $\gamma \in [0.5, 1]$.

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arXiv2023

Nearly Equitable Allocations Beyond Additivity and Monotonicity

Equitability (EQ) in fair division requires that items be allocated such that all agents value the bundle they receive equally. With indivisible items, an equitable allocation may not exist, and hence we instead consider a meaningful analog, EQx, that requires equitability up to any item. EQx allocations exist for monotone, additive valuations. However, if (1) the agents' valuations are not additive or (2) the set of indivisible items includes both goods and chores (positively and negatively valued items), then prior to the current work it was not known whether EQx allocations exist or not. We study both the existence and efficient computation of EQx allocations. (1) For monotone valuations (not necessarily additive), we show that EQx allocations always exist. Also, for the large class of weakly well-layered valuations, EQx allocations can be found in polynomial time. Further, we prove that approximately EQx allocations can be computed efficiently under general monotone valuations. (2) For non-monotone valuations, we show that an EQx allocation may not exist, even for two agents with additive valuations. Under some special cases, however, we establish existence and efficient computability of EQx allocations. This includes the case of two agents with additive valuations where each item is either a good or a chore, and there are no mixed items. In addition, we show that, under nonmonotone valuations, determining the existence of EQx allocations is weakly NP-hard for two agents and strongly NP-hard for more agents.

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arXiv2023

Nash Regret Guarantees for Linear Bandits

We obtain essentially tight upper bounds for a strengthened notion of regret in the stochastic linear bandits framework. The strengthening -- referred to as Nash regret -- is defined as the difference between the (a priori unknown) optimum and the geometric mean of expected rewards accumulated by the linear bandit algorithm. Since the geometric mean corresponds to the well-studied Nash social welfare (NSW) function, this formulation quantifies the performance of a bandit algorithm as the collective welfare it generates across rounds. NSW is known to satisfy fairness axioms and, hence, an upper bound on Nash regret provides a principled fairness guarantee. We consider the stochastic linear bandits problem over a horizon of $T$ rounds and with set of arms ${X}$ in ambient dimension $d$. Furthermore, we focus on settings in which the stochastic reward -- associated with each arm in ${X}$ -- is a non-negative, $ν$-sub-Poisson random variable. For this setting, we develop an algorithm that achieves a Nash regret of $O\left( \sqrt{\frac{dν}{T}} \log( T |X|)\right)$. In addition, addressing linear bandit instances in which the set of arms ${X}$ is not necessarily finite, we obtain a Nash regret upper bound of $O\left( \frac{d^\frac{5}{4}ν^{\frac{1}{2}}}{\sqrt{T}} \log(T)\right)$. Since bounded random variables are sub-Poisson, these results hold for bounded, positive rewards. Our linear bandit algorithm is built upon the successive elimination method with novel technical insights, including tailored concentration bounds and the use of sampling via John ellipsoid in conjunction with the Kiefer-Wolfowitz optimal design.

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arXiv2023

Learning Good Interventions in Causal Graphs via Covering

We study the causal bandit problem that entails identifying a near-optimal intervention from a specified set $A$ of (possibly non-atomic) interventions over a given causal graph. Here, an optimal intervention in ${A}$ is one that maximizes the expected value for a designated reward variable in the graph, and we use the standard notion of simple regret to quantify near optimality. Considering Bernoulli random variables and for causal graphs on $N$ vertices with constant in-degree, prior work has achieved a worst case guarantee of $\widetilde{O} (N/\sqrt{T})$ for simple regret. The current work utilizes the idea of covering interventions (which are not necessarily contained within ${A}$) and establishes a simple regret guarantee of $\widetilde{O}(\sqrt{N/T})$. Notably, and in contrast to prior work, our simple regret bound depends only on explicit parameters of the problem instance. We also go beyond prior work and achieve a simple regret guarantee for causal graphs with unobserved variables. Further, we perform experiments to show improvements over baselines in this setting.

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arXiv2023

Guaranteeing Envy-Freeness under Generalized Assignment Constraints

We study fair division of goods under the broad class of generalized assignment constraints. In this constraint framework, the sizes and values of the goods are agent-specific, and one needs to allocate the goods among the agents fairly while further ensuring that each agent receives a bundle of total size at most the corresponding budget of the agent. Since, in such a constraint setting, it may not always be feasible to partition all the goods among the agents, we conform -- as in recent works -- to the construct of charity to designate the set of unassigned goods. For this allocation framework, we obtain existential and computational guarantees for envy-free (appropriately defined) allocation of divisible and indivisible goods, respectively, among agents with individual, additive valuations for the goods. We deem allocations to be fair by evaluating envy only with respect to feasible subsets. In particular, an allocation is said to be feasibly envy-free (FEF) iff each agent prefers its bundle over every (budget) feasible subset within any other agent's bundle (and within the charity). The current work establishes that, for divisible goods, FEF allocations are guaranteed to exist and can be computed efficiently under generalized assignment constraints. In the context of indivisible goods, FEF allocations do not necessarily exist, and hence, we consider the fairness notion of feasible envy-freeness up to any good (FEFx). We show that, under generalized assignment constraints, an FEFx allocation of indivisible goods always exists. In fact, our FEFx result resolves open problems posed in prior works. Further, for indivisible goods and under generalized assignment constraints, we provide a pseudo-polynomial time algorithm for computing FEFx allocations, and a fully polynomial-time approximation scheme (FPTAS) for computing approximate FEFx allocations.

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