arXiv subjects

Shiji Xing

Publications and source records attributed to Shiji Xing.

3 recordsLinked to original sources

arXiv2025

Constant Weighted Maximin Share Approximations for Chores

We study the fair allocation of indivisible chores among agents with asymmetric weights. Among the various fairness notions, weighted maximin share (WMMS) stands out as particularly compelling. Despite its appeal, the existence of a constant-factor approximation for WMMS has remained an important open problem in weighted fair division [Aziz et al., 2022, Suksompong, 2025]. Prior to our work, the best known approximation ratio was $O(\log n)$, where $n$ is the number of agents. In this paper, we make significant progress by presenting the first constant-factor approximation algorithm for WMMS. Our main contributions are as follows: [1] We design the first algorithm that guarantees a 12-approximate WMMS allocation, substantially improving upon the previous $O(\log n)$ upper bound. Our approach introduces a novel analytical framework based on canonical instance reductions, agent delegation, and proxy cost functions to effectively bound agents' costs. Additionally, we provide a polynomial-time implementation for any approximate WMMS algorithm, incurring a factor of 2 loss in the approximation ratio. [2] We present an improved worst-case lower bound, showing that no algorithm can achieve better than 2-approximate WMMS, thereby strengthening the previous best lower bound of 1.366. We further construct a general hard instance, which provides lower bounds for an arbitrary number of agents. [3] Beyond worst-case bounds, we precisely characterize the optimal approximation ratio curve for every possible weight distribution in the two-agent case. Notably, our results imply that a WMMS allocation may not exist for any two agents with different weights, in sharp contrast to the symmetric case where an MMS allocation always exists.

cs.GT↗

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 $\Theta(n)$ 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.

cs.GT↗

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.

cs.GT↗