Comparison-Based Fair Division of Indivisible Chores
We investigate the query complexity of fairly allocating $m$ indivisible chores among $n$ agents with additive cost functions. We depart from the standard cardinal model and assume only comparison access: an algorithm may ask an agent which of two bundles is less costly, but never observes numerical costs. Our first results concern proportionality up to one item (PROP1). We design comparison-based algorithms that compute PROP1 allocations using $O(n^3\log m)$ comparison queries. When the chores are arranged in a fixed order and allocations are required to be contiguous, we compute a contiguous PROP1 allocation using $O(n^3 \log^2 m)$ comparison queries. Our main result concerns the maximin share (MMS) guarantee. We show that for any fixed number of agents $n$ and constant $\varepsilon>0$, a $\left(13/11 +\varepsilon\right)$-MMS allocation can be computed with a comparison complexity logarithmic in $m$. Remarkably, comparison access suffices to match the state-of-the-art $13/11$ cardinal-access guarantee of Huang and Segal-Halevi up to an arbitrarily small loss. Furthermore, our result implies that the MMS distortion of comparison access (i.e., the worst-case multiplicative loss in MMS fairness incurred by observing only comparisons rather than numerical costs) is at most $13/11$. Finally, we show that, for three agents, an allocation satisfying envy-freeness up to one item (EF1) can be computed using $O(\log m)$ comparison queries.