Computing Efficient Envy-Free Partial Allocations of Indivisible Goods

TL;DR

The paper investigates envy-free partial allocations (EF-PA) of indivisible goods under mild efficiency constraints, formalizing the problem with an efficiency threshold $t$ across utilitarian and egalitarian welfare as well as allocation size and min-cardinality. It delivers a comprehensive complexity map: for binary utilities, all EF-PA variants are solvable in polynomial time when $t=1$, with several variants remaining tractable or FPT in $t$ for larger thresholds; in contrast, all four variants become strongly NP-hard under ternary valuations even at $t=1$, via reductions from 3-partition and X3C and elaborate gadget constructions. A central algorithmic tool is the use of envy-free matchings (EFM) and the EFM partition, which yields efficient solutions for certain goals (notably esw via maximum bipartite matching) and structural insights for other goals (usw, size, mcar) in the binary setting. Overall, the work clarifies the computational boundaries between tractable and intractable fair allocations under partial allocation regimes, highlighting practical algorithms for binary preferences while identifying hard regimes for richer value structures.

Abstract

Envy-freeness is one of the most prominent fairness concepts in the allocation of indivisible goods. Even though trivial envy-free allocations always exist, rich literature shows this is not true when one additionally requires some efficiency concept (e.g., completeness, Pareto-efficiency, or social welfare maximization). In fact, in such case even deciding the existence of an efficient envy-free allocation is notoriously computationally hard. In this paper, we explore the limits of efficient computability by relaxing standard efficiency concepts and analyzing how this impacts the computational complexity of the respective problems. Specifically, we allow partial allocations (where not all goods are allocated) and impose only very mild efficiency constraints, such as ensuring each agent receives a bundle with positive utility. Surprisingly, even such seemingly weak efficiency requirements lead to a diverse computational complexity landscape. We identify several polynomial-time solvable or fixed-parameter tractable cases for binary utilities, yet we also find NP-hardness in very restricted scenarios involving ternary utilities.

Theorems & Definitions (28)

• Theorem 1
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• Theorem 2
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• Theorem 3: aigner2022
• Lemma 1
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• Theorem 4
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• Theorem 5