The Cube-Root Phenomenon in Online Carpooling
We consider the online carpooling problem, where edges arrive online and must be oriented immediately while keeping the discrepancy between the indegree and outdegree at each vertex small. We prove that the natural Greedy algorithm incurs discrepancy $O(\min\{T^{1/3},n\})$ after $T$ arrivals. This resolves a question of Ajtai et al., who showed that any deterministic algorithm must incur $Ω(\min\{T^{1/3},n\})$ discrepancy, and gave an algorithm with $O(\min\{T^{1/2},n\})$ discrepancy. We also show a similar square-root to cube-root improvement in the stochastic setting, where $O(n)$ edges are sampled independently from an underlying $n$-vertex graph $G$. Formally, we show an $O((\log n)^{1/3})$ bound for random arrivals from any $Δ$-regular graph $G$. When $Δ= Ω((\log n)^3)$, we show the more refined bound of $O((\log n/\log Δ)^{1/3}+\log\log n)$ on the discrepancy. We show that the cube-root term in the previous bound is essential, while the $\log\log n$ term is already known to be necessary for random arrivals from complete graphs. The previous upper bounds here were $O((\log n)^{1/2})$, which follow from the breakthrough works on online discrepancy due to Kulkarni, Reis, and Rothvoss, and Aden-Ali. Our techniques for proving such cube-root-type bounds may be of independent interest, as the standard quadratic-potential and subgaussian analyses underlying the previous general bounds appear inherently unable to go below square-root-type guarantees.