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

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.