arXiv subjects

Ankang Sun

Publications and source records attributed to Ankang Sun.

16 recordsLinked to original sources

arXiv2026

Toward Constant-Factor Maximin Share Guarantees for Subadditive Valuations

For $n$ agents and $m$ indivisible goods, an agent's maximin share (MMS) is the largest minimum bundle value she can obtain by partitioning the goods into $n$ bundles. Constant MMS approximations are known for additive, submodular, or XOS valuations, but whether such a guarantee exists for general subadditive valuations remains open. We advance this question in two directions. For general subadditive valuations, we exponentially improve the guarantee from $Ω(1/\log\log n)$ to $Ω(1/\log^* n)$, where $\log^* n$ is the iterated logarithm. We then seek constant MMS guarantees under additional structure on subadditive valuations by studying $q$-partitioning valuations: every partition of a bundle into at most $q$ groups admits nonnegative group prices that sum to the bundle's value and whose sum on any collection of groups is at most the value of their union. This hierarchy ranges from subadditivity at $q=2$ to XOS for $q\ge m$. We give the first study of MMS in the intermediate classes and prove constant guarantees whenever $q/n$ has a fixed positive lower bound, including $1/6$ for $q\ge n$ and $1/4$ for $q\ge2n$. For XOS valuations, we further improve the universal guarantee from $4/17\approx 0.2353$ to $5/16=0.3125$. For subadditive valuations, we prove a bidding amplification theorem that gives an exponential bound on each agent's probability of receiving less than half her MMS, while satisfying her budget in every outcome. For $q$-partitioning valuations, we give a scalar criterion that converts bounds on the value lost from disjoint deletions within a bundle into bounds on potential losses, even when an agent contributes zero below the target bundle value. For XOS valuations, we prove a reservation theorem that selects goods for integral assignment together with auxiliary additive valuations and accounts for the loss in rounding.

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arXiv2026

Settling PROPm and PROPavg in Graphical Resource Allocation

We study proportional fairness in graphical resource allocation, where agents are vertices, indivisible items are edges, and each item must be allocated to one of its two endpoints. It has been proved that PROP1 orientations always exist and PROPx orientations may not, but it has remained open whether the intermediate relaxations PROPm and PROPavg (both can be satisfied without graphical constraints) can always be satisfied. In this paper, we resolve this gap. We prove that PROPm orientations always exist for goods on multigraphs and can be computed efficiently. The guarantee extends to chores and a mixture of goods and chores. In sharp contrast, we show that PROPavg orientations need not exist, even for simple graphs with binary valuations, and that deciding their existence is NP-complete. We further quantify the efficiency loss of PROPm: for goods, the price of PROPm is exactly $2$, which is one of the few settings where a constant bound on the price of fairness can be obtained. We complement these results with hardness results for welfare optimization under PROPm and for the existence of orientations that are simultaneously PROPm and Pareto optimal.

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arXiv2026

Connected EF1 Allocations Exist in Discrete Chore Cutting

In this paper, we prove the existence of an envy-free up to one item (EF1) division for a discrete chore. Our approach builds on the powerful framework of Simmons-Su, which leverages Sperner's lemma to guarantee the existence of a simplex corresponding to a sequence of similar fractional divisions, ensuring that each agent is satisfied with a different bundle. Bilò et al. [2022] introduced a rounding technique that converts the fractional divisions into a connected integral EF1 division for goods when there are at most four agents, and this method was later extended by Igarashi [2023] to accommodate any number of agents. However, these rounding techniques for goods do not directly apply to chores because the definitions of EF1 differ in the two settings. To overcome this asymmetry, we modify the existing rounding techniques and show that connected EF1 divisions exist for a discrete chore.

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arXiv2026

Online Fair Division: Pushing the Frontier of Approximate Proportionality

Online fair division captures allocation problems in which indivisible resources arrive over time and must be assigned before future resources are known. Understanding what fairness remains achievable when allocation decisions are immediate and irrevocable is a fundamental question in this setting. We study deterministic online allocation among $n$ agents with nonnegative additive valuations, where the number of goods is unknown and the adversary can adapt to previous allocation decisions. We focus on proportionality up to one good (PROP1) and examine how advance information affects the achievable guarantees. Without additional future information, we give a deterministic algorithm that guarantees $Ω(\frac{1}{\log(nm)})$-PROP1 after every round, where $m$ is the number of goods at termination. We complement this result by showing that, for every $n$ and sufficiently large $m$, every deterministic algorithm has an adaptive instance with $m$ goods on which its allocation has PROP1 approximation guarantee $O(\frac{\log \log m}{\log m})$. Thus, when the number of agents is fixed, our upper and lower bounds on the competitive ratio differ by at most an $O(\log\log m)$ factor. These results answer an open question proposed by Choo et al. on whether a nontrivial deterministic approximation for PROP1 can be obtained. We also study the setting where the algorithm knows the predictions of the maximum item value for every agent. When the predictions are accurate, we give a deterministic $\frac{1}{2}$-PROP1 algorithm, improving the $\frac{1}{n}$ guarantee of Choo et al. to a constant. We further establish an explicit upper bound below one on the competitive ratio, even for two agents with accurate predictions. Finally, we give a single deterministic algorithm that guarantees $\frac{1}{2}$-PROP1 when predictions are accurate and $Ω(\frac{1}{\log (nm)})$-PROP1 for arbitrary predictions.

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arXiv2026

Fair Orientations: Proportionality and Equitability

We study the fair allocation of indivisible items under relevance constraints, where each agent has a set of relevant items and can only receive items that are relevant to them. While the relevance constraint has been studied in recent years, existing work has largely focused on envy-freeness. Our work extends this study to other key fairness criteria -- such as proportionality, equitability, and their relaxations -- in settings where the items may be goods, chores, or a mixture of both. We complement the literature by presenting a picture of the existence and computational complexity of the considered criteria.

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arXiv2026

Multi-Agent Non-Discriminatory Contracts

We study multi-agent contracts, in which a principal delegates a task to multiple agents and incentivizes them to exert effort. Prior research has mostly focused on maximizing the principal's utility, often resulting in highly disparate payments among agents. Such disparities among agents may be undesirable in practice, for example, in standardized public contracting or worker cooperatives where fairness concerns are essential. Motivated by these considerations, our objective is to quantify the tradeoff between maximizing the principal's utility and equalizing payments among agents, which we call the price of non-discrimination. Our first result is an almost tight bound on the price of non-discrimination, which scales logarithmically with the number of agents. This bound can be improved to a constant by allowing some relaxation of the non-discrimination requirement. We then provide a comprehensive characterization of the tradeoff between the level of non-discrimination and the loss in the optimal utility.

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arXiv2025

Bin Packing and Covering: Pushing the Frontier on the Maximin Share Fairness

We study a fundamental fair allocation problem, where the agent's value is determined by the number of bins either used to pack or cover the items allocated to them. Fairness is evaluated using the maximin share (MMS) criterion. This problem is not only motivated by practical applications, but also serves as a natural framework for studying group fairness. As MMS is not always satisfiable, we consider two types of approximations: cardinal and ordinal. For cardinal approximation, we relax the requirements of being packed or covered for a bin, and for ordinal approximation, we relax the number of bins that are packed or covered. For all models of interest, we provide constant approximation algorithms.

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arXiv2025

On the Subsidy of Envy-Free Orientations in Graphs

We study a fair division problem in (multi)graphs where $n$ agents (vertices) are pairwise connected by items (edges), and each agent is only interested in its incident items. We consider how to allocate items to incident agents in an envy-free manner, i.e., envy-free orientations, while minimizing the overall payment, i.e., subsidy. We first prove that computing an envy-free orientation with the minimum subsidy is NP-hard, even when the graph is simple and the agents have bi-valued additive valuations. We then bound the worst-case subsidy. We prove that for any multigraph (i.e., allowing parallel edges) and monotone valuations where the marginal value of each good is at most \$1 for each agent, \$1 each (a total subsidy of $n-1$, where $n$ is the number of agents) is sufficient. This is one of the few cases where linear subsidy $Θ(n)$ is known to be necessary and sufficient to guarantee envy-freeness when agents have monotone valuations. When the valuations are additive (while the graph may contain parallel edges) and when the graph is simple (while the valuations may be monotone), we improve the bound to $n/2$ and $n-2$, respectively. Moreover, these two bounds are tight.

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arXiv2025

The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective

We adopt a parametric approach to analyze the worst-case degradation in social welfare when the allocation of indivisible goods is constrained to be fair. Specifically, we are concerned with cardinality-constrained allocations, which require that each agent has at most $k$ items in their allocated bundle. We propose the notion of the price of cardinality, which captures the worst-case multiplicative loss of utilitarian or egalitarian social welfare resulting from imposing the cardinality constraint. We then characterize tight or almost-tight bounds on the price of cardinality as exact functions of the instance parameters, demonstrating how the social welfare improves as $k$ is increased. In particular, one of our main results refines and generalizes the existing asymptotic bound on the price of balancedness, as studied by Bei et al. [BLMS21]. We also further extend our analysis to the problem where the items are partitioned into disjoint categories, and each category has its own cardinality constraint. Through a parametric study of the price of cardinality, we provide a framework which aids decision makers in choosing an ideal level of cardinality-based fairness, using their knowledge of the potential loss of utilitarian and egalitarian social welfare.

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arXiv2024

A Fair Allocation is Approximately Optimal for Indivisible Chores, or Is It?

In this paper, we study the allocation of indivisible chores and consider the problem of finding a fair allocation that is approximately efficient. We shift our attention from the multiplicative approximation to the additive one. Our results are twofold, with (1) bounding how the optimal social cost escalates resulting from fairness requirements and (2) presenting the hardness of approximation for the problems of finding fair allocations with the minimum social cost. To quantify the escalation, we introduce cost of fairness (CoF) $\unicode{x2014}$ an alternative to the price of fairness (PoF) $\unicode{x2014}$ to bound the difference (v.s. ratio for PoF) between the optimal social cost with and without fairness constraints in the worst-case instance. We find that CoF is more informative than PoF for chores in the sense that the PoF is infinity regarding all EQX (equitable up to any item), EQ1 (equitable up to one item) and EF1 (envy-free up to one item), while the CoF is $n$ regarding EQX and 1 regarding EQ1 and EF1, where $n$ is the number of agents. For inapproximability, we present a detailed picture of hardness of approximation. We prove that finding the optimal EQX allocation within an additive approximation factor of $n$ is NP-hard for any $n \geq 2$ where $n$ is the number of agents and the cost functions are normalized to 1. For EQ1 and EF1, the problem is NP-hard when the additive factor is a constant and $n \geq 3$. When $n = 2$, we design additive approximation schemes for EQ1 and EF1.

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arXiv2024

Randomized Strategyproof Mechanisms with Best of Both Worlds Fairness and Efficiency

We study the problem of mechanism design for allocating a set of indivisible items among agents with private preferences on items. We are interested in such a mechanism that is strategyproof (where agents' best strategy is to report their true preferences) and is expected to ensure fairness and efficiency to a certain degree. We first present an impossibility result that a deterministic mechanism does not exist that is strategyproof, fair and efficient for allocating indivisible chores. We then utilize randomness to overcome the strong impossibility. For allocating indivisible chores, we propose a randomized mechanism that is strategyproof in expectation as well as ex-ante and ex-post (best of both worlds) fair and efficient. For allocating mixed items, where an item can be a good (i.e., with a positive utility) for one agent but a chore (i.e., a with negative utility) for another, we propose a randomized mechanism that is strategyproof in expectation with best of both worlds fairness and efficiency when there are two agents.

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arXiv2023

Envy-free House Allocation under Uncertain Preferences

We study the envy-free house allocation problem when agents have uncertain preferences over items and consider several well-studied preference uncertainty models. The central problem that we focus on is computing an allocation that has the highest probability of being envy-free. We show that each model leads to a distinct set of algorithmic and complexity results, including detailed results on (in-)approximability. En route, we consider two related problems of checking whether there exists an allocation that is possibly or necessarily envy-free. We give a complete picture of the computational complexity of these two problems for all the uncertainty models we consider.

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arXiv2023

On Hill's Worst-Case Guarantee for Indivisible Bads

When allocating objects among agents with equal rights, people often evaluate the fairness of an allocation rule by comparing their received utilities to a benchmark share - a function only of her own valuation and the number of agents. This share is called a guarantee if for any profile of valuations there is an allocation ensuring the share of every agent. When the objects are indivisible goods, Budish [J. Political Econ., 2011] proposed MaxMinShare, i.e., the least utility of a bundle in the best partition of the objects, which is unfortunately not a guarantee. Instead, an earlier pioneering work by Hill [Ann. Probab., 1987] proposed for a share the worst-case MaxMinShare over all valuations with the same largest possible single-object value. Although Hill's share is more conservative than the MaxMinShare, it is an actual guarantee and its computation is elementary, unlike that of the MaxMinShare which involves solving an NP-hard problem. We apply Hill's approach to the allocation of indivisible bads (objects with disutilities or costs), and characterise the tight closed form of the worst-case MinMaxShare for a given value of the worst bad. We argue that Hill's share for allocating bads is effective in the sense of being close to the original MinMaxShare value, and there is much to learn about the guarantee an agent can be offered from the disutility of her worst single bad. Furthermore, we prove that the monotonic cover of Hill's share is the best guarantee that can be achieved in Hill's model for all allocation instances.

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arXiv2022

On the Price of Fairness of Allocating Contiguous Blocks

In this work, we revisit the problem of fairly allocating a number of indivisible items that are located on a line to multiple agents. A feasible allocation requires that the allocated items to each agent are connected on the line. The items can be goods on which agents have non-negative utilities, or chores on which the utilities are non-positive. Our objective is to understand the extent to which welfare is inevitably sacrificed by enforcing the allocations to be fair, i.e., price of fairness (PoF). We study both egalitarian and utilitarian welfare. Previous works by Suksompong [Discret. Appl. Math., 2019] and Höhne and van Stee [Inf. Comput., 2021] have studied PoF regarding the notions of envy-freeness and proportionality. However, these fair allocations barely exist for indivisible items, and thus in this work, we focus on the relaxations of maximin share fairness and proportionality up to one item, which are guaranteed to be satisfiable. For most settings, we give (almost) tight ratios of PoF and all the upper bounds are proved by designing polynomial time algorithms.

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arXiv2022

Approximate Group Fairness for Clustering

We incorporate group fairness into the algorithmic centroid clustering problem, where $k$ centers are to be located to serve $n$ agents distributed in a metric space. We refine the notion of proportional fairness proposed in [Chen et al., ICML 2019] as {\em core fairness}, and $k$-clustering is in the core if no coalition containing at least $n/k$ agents can strictly decrease their total distance by deviating to a new center together. Our solution concept is motivated by the situation where agents are able to coordinate and utilities are transferable. A string of existence, hardness and approximability results is provided. Particularly, we propose two dimensions to relax core requirements: one is on the degree of distance improvement, and the other is on the size of deviating coalition. For both relaxations and their combination, we study the extent to which relaxed core fairness can be satisfied in metric spaces including line, tree and general metric space, and design approximation algorithms accordingly.

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arXiv2021

Fairness Criteria for Allocating Indivisible Chores: Connections and Efficiencies

We study several fairness notions in allocating indivisible chores (i.e., items with non-positive values) to agents who have additive and submodular cost functions. The fairness criteria we are concern with are envy-free up to any item (EFX), envy-free up to one item (EF1), maximin share (MMS), and pairwise maximin share (PMMS), which are proposed as relaxations of envy-freeness in the setting of additive cost functions. For allocations under each fairness criterion, we establish their approximation guarantee for other fairness criteria. Under the additive setting, our results show strong connections between these fairness criteria and, at the same time, reveal intrinsic differences between goods allocation and chores allocation. However, such strong relationships cannot be inherited by the submodular setting, under which PMMS and MMS are no longer relaxations of envy-freeness and, even worse, few non-trivial guarantees exist. We also investigate efficiency loss under these fairness constraints and establish their prices of fairness.

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