Maximizing $p$-Mean Social Welfare in the High-Multiplicity Setting: Few Agent Types and Few Item Types

Trung Thanh Nguyen, Khaled Elbassioni

The $p$-mean welfare objective unifies several classical social welfare criteria for the allocation of indivisible goods. We study its maximization under additive nonnegative utilities when item and agent multiplicities are encoded in binary. For every fixed finite rational $p<1$, $p\neq0$, we show that the problem is $\mathsf{NP}$-hard with only two item types. Nash welfare ($p=0$) and egalitarian welfare ($p=-\infty$) are NP-hard with three item types. These results hold both for computing an optimal allocation and for rational-threshold decision, and the utilities and threshold can be required to be positive integers. We also show that maximizing $p$-mean social welfare is strongly NP-hard with one agent type and an unrestricted number of item types, for every fixed finite rational $p<1$ and for $p=-\infty$. A quantitative gap in this reduction rules out an FPTAS in the latter setting unless $\mathsf{P}=\mathsf{NP}$. On the positive side, for a fixed number of item types and an arbitrary number of agent types, we give an FPTAS for every fixed $p\in\mathbb Q\cup\{-\infty\}$. Its running time is polynomial in the compact input length and in $1/\varepsilon$, and it returns a…