Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.03318

Constant-Factor EFX Exists for Chores

Jugal Garg·Aniket Murhekar·John Qin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the problem of fair allocation of chores to 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. The current best guarantee is the existence of $O(n^2)$-EFX allocations for $n$ agents, obtained through a sophisticated algorithm (Zhou and Wu, 2022). In this paper, we show the existence of $4$-EFX allocations, providing the first constant-factor approximation of EFX. We also 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, thus improving the current best factor of $O(n)$-EFX without any efficiency guarantees. For general additive instances, the existence of allocations that are $\alpha$-EF$k$ and PO has remained open for any constant values of $\alpha$ 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 $2$-EF$2$ and PO. Our results are obtained via a novel economic framework called earning restricted (ER) competitive equilibrium for fractional allocations, which limits agents' earnings from each chore. We show the existence of ER equilibria by formulating it as an linear complementarity problem (LCP) and proving that the classic complementary pivot algorithm on the LCP terminates at an ER equilibrium. We design algorithms that carefully round fractional ER equilibria, and perform bundle swaps and merges to meet the desired fairness and efficiency criteria. We expect that the concept of ER equilibrium will be useful in deriving further results on related problems.

Explore related subjects

cs.GT

Keep this discovery

BibTeXRIS

Jugal Garg, Aniket Murhekar, John Qin. 2024-07-03. Constant-Factor EFX Exists for Chores. https://arxiv.org/abs/2407.03318

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

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

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗
arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗