Fair and Efficient Allocations: Decision Problems in the Gap of Polynomial Hierarchy

2026-09-25

AI Summary

This study addresses the long-standing open problem posed by Bouveret and Lang regarding the computational complexity of simultaneously satisfying envy-freeness and efficiency in the fair allocation of indivisible items. By employing fine-grained complexity analysis and polynomial hierarchy theory, combined with Pareto optimality and social welfare optimization modeling, this work systematically characterizes the computational complexity across varying numbers of agents and valuation models. The primary contribution lies in establishing the Σ2p-completeness of this decision problem and providing an intermediate-class classification. It reveals that numerous variants fall into the gap between NP and coNP within the polynomial hierarchy, thereby filling critical theoretical voids. Furthermore, this research generalizes the prior results obtained by De Keijzer et al. under restricted settings to the general case.

Abstract

We consider the fair division problem with indivisible goods and study the following decision problem: given a fair division instance, does there exist an allocation that is envy-free and efficient? We consider two efficiency criteria: Pareto-optimality and social welfare optimality. We provide a complete landscape on the computational complexity of this decision problem, with the number of agents ranging from $2$ to $\infty$, both additive valuations and general valuations, and the more restricted class of $k$-ary valuation functions (where an item's marginal value is restricted to $\{0,1,\ldots,k-1\}$ for some constant $k\geq2$). One interesting observation is that many versions of the above-mentioned decision problems fall into the ``gap''between the first and the second levels of the polynomial hierarchy. Specifically, assuming the polynomial hierarchy does not collapse to the first level (i.e., assuming $\text{NP}\neq\text{coNP}$), these problems are in $(\Sigma_2^{\text{p}}\cap\Pi_2^{\text{p}})\setminus(\text{NP}\cup\text{coNP})$. In particular, we provide a fine-grained complexity analysis across different parameter regimes, including the number of agents and the choice of valuation models. Depending on different parameters, many problems admit different complexity classifications, ranging from the intermediate classes $\Theta_2^{\text{p}}$ and $\Delta_2^{\text{p}}$ between the two levels to $\Sigma_2^{\text{p}}$-completeness. Finally, De Keijzer et al. show the $\Sigma_2^{\text{p}}$-completeness of the decision problem when considering Pareto-optimality as the efficiency criterion with additive valuations. Our main results extend this result to more restricted settings, such as instances with a constant number of agents or $3$-ary valuation functions, which resolves the open problem given by Bouveret and Lang.

Problem

• fair division
• indivisible goods
• computational complexity
• polynomial hierarchy
• envy-free allocation

Innovation

• Fair Division
• Computational Complexity
• Polynomial Hierarchy
• Pareto-optimality
• Indivisible Goods