arXiv subjects

Zehan Lin

Publications and source records attributed to Zehan Lin.

7 recordsLinked to original sources

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗

arXiv2026

Non-Existence of PMMS Allocations and a $4/3$-PMMS Guarantee for Additive Chores

We study pairwise maximin share (PMMS) fairness for indivisible items with additive preferences. We give a polynomial-time reduction from chores to goods that preserves the existence of a PMMS allocation. Together with known nonexistence results for chores, this yields nonexistence for additive goods. In addition, we show that deciding if a given instance admits a PMMS allocation is NP-hard. We also give explicit instances whose PMMS factors are $226/227$ for goods and $1.102065$ for chores, certified by exact enumeration. Complementing these impossibility results, we prove that every additive-chore instance admits a $4/3$-PMMS allocation.

cs.GT↗

arXiv2026

Comparison-Based Fair Division of Indivisible Chores

We investigate the query complexity of fairly allocating $m$ indivisible chores among $n$ agents with additive cost functions. We depart from the standard cardinal model and assume only comparison access: an algorithm may ask an agent which of two bundles is less costly, but never observes numerical costs. Our first results concern proportionality up to one item (PROP1). We design comparison-based algorithms that compute PROP1 allocations using $O(n^3\log m)$ comparison queries. When the chores are arranged in a fixed order and allocations are required to be contiguous, we compute a contiguous PROP1 allocation using $O(n^3 \log^2 m)$ comparison queries. Our main result concerns the maximin share (MMS) guarantee. We show that for any fixed number of agents $n$ and constant $\varepsilon>0$, a $\left(13/11 +\varepsilon\right)$-MMS allocation can be computed with a comparison complexity logarithmic in $m$. Remarkably, comparison access suffices to match the state-of-the-art $13/11$ cardinal-access guarantee of Huang and Segal-Halevi up to an arbitrarily small loss. Furthermore, our result implies that the MMS distortion of comparison access (i.e., the worst-case multiplicative loss in MMS fairness incurred by observing only comparisons rather than numerical costs) is at most $13/11$. Finally, we show that, for three agents, an allocation satisfying envy-freeness up to one item (EF1) can be computed using $O(\log m)$ comparison queries.

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arXiv2026

Almost Envy-Freeness for Additive Mixed Manna with Entitlements: Deterministic and Randomized Guarantees

We investigate the fair allocation of indivisible items among agents with asymmetric entitlements in mixed manna settings, where the items consist of both goods and chores. For additive valuations, we establish that weighted envy-free up to one item (WEF1) allocations always exist and can be computed in polynomial time. We also study fair and efficient allocation and show that weighted envy-freeness up to one transfer (WEF1T) is compatible with fractional Pareto optimality (fPO) for every mixed-manna instance. This relaxation from WEF1 to WEF1T is tight, as demonstrated by our impossibility result. We further show a best-of-both-worlds result via a finite lottery that guarantees weighted envy-freeness (WEF) in expectation, with every realized allocation satisfying WEF1T and achieving the tight characterization complemented by the existing impossibility result.

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arXiv2026

Non-Existence of EFX Chore Allocations for Monotone Cost Functions with Binary Marginals

We study the existence of envy-free up to any item (EFX) allocations of indivisible chores when agents have monotone cost functions with binary marginals. For indivisible goods, the corresponding existence question is known to have an affirmative answer for general monotone functions with binary marginals. For chores, however, the existence of EFX allocation was previously known only for more restricted classes, while the general binary-marginal case remained unresolved. In this paper, we provide two counterexamples based on the same 18-agent, 53-chore word gadget, with one cost profile for binary XOS costs and another for binary supermodular costs. In both cases, a complete EFX allocation need not exist. Finally, we formalize and verify our main results in Lean 4.

cs.GT↗

arXiv2026

Allocating Chores with Restricted Additive Costs: Achieving EFX, MMS, and Efficiency Simultaneously

In a web-based review platform, papers from various research fields must be assigned to a group of reviewers. Each paper has an inherent cost, which represents the effort required for reading and evaluating it (e.g., the paper's length). Reviewers can bid on papers they are interested in, and if they are assigned a paper they have bid on, no cost is incurred. Otherwise, the inherent cost $c(e)$ for paper $e$ applies. We capture this with a model of restricted additive costs: every item $e$ has a cost $c(e)$, and each agent either incurs $0$ or $c(e)$ for $e$. In this work, we study how to allocate such chores fairly and efficiently. We propose an algorithm for computing allocations that are both EFX and MMS. Furthermore, we show that our algorithm achieves a $2$-approximation of the optimal social cost, and the approximation ratio is optimal. We also show that slightly weaker fairness guarantees can be obtained if one requires the algorithm to run in polynomial time.

cs.GT↗

arXiv2025

Approximately EFX and PO Allocations for Bivalued Chores

We consider the computation for allocations of indivisible chores that are approximately EFX and Pareto optimal (PO). Recently, Garg et al. (2024) show the existence of $3$-EFX and PO allocations for bi-valued instances, where the cost of an item to an agent is either $1$ or $k$ (where $k > 1$) by rounding the (fractional) earning restricted equilibrium. In this work, we improve the approximation ratio to $(2-1/k)$, while preserving the Pareto optimality. Instead of rounding fractional equilibrium, our algorithm starts with the integral EF1 equilibrium for bi-valued chores, introduced by Garg et al. (AAAI 2022) and Wu et al. (EC 2023), and reallocates items until approximate EFX is achieved. We further improve our result for the case when $k=2$ and devise an algorithm that computes EFX and PO allocations.

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