Logarithmic Comparison-Based Query Complexity for Fair Division of Indivisible Goods

We study the problem of fairly allocating $m$ indivisible goods to $n$ agents, where agents may have different preferences over the goods. In the traditional setting, agents' valuations are provided as inputs to the algorithm. In this paper, we adopt the query model, which has been widely considered for other similar problems (such as matching [Nis21], graph isomorphism [OS18], and equilibrium in game [Bab16]), and apply it to the fair division problem. In particular, we consider a new \emph{comparison-based query model}, where the algorithm presents two bundles of goods to an agent and the agent responds by telling the algorithm which bundle she prefers. We investigate the query complexity for computing allocations with several fairness notions, including \emph{proportionality up to one good} (PROP1), \emph{envy-freeness up to one good} (EF1), and \emph{maximin share} (MMS). Our main result is an algorithm that computes an allocation that satisfies both PROP1 and $\frac12$-MMS within $O(\log m)$ queries with a constant number of $n$ agents. For identical and additive valuation, we present an algorithm for computing an EF1 allocation within $O(\log m)$ queries with a constant number of $n$ agents. To complement the positive results, we show that the lower bound of the query complexity for any of the three fairness notions is $\Omega(\log m)$ even with two agents.