Discrepancy for Random Arrivals from Arbitrary Graphs

Establish that, for edges sampled independently and uniformly from an arbitrary, possibly non-regular, graph, there exists an online orientation algorithm whose discrepancy is O((log T)^{1/3}) after T arrivals.

Background

The paper proves an O((log n){1/3}) discrepancy bound for O(n) independent uniform edge samples from any Δ-regular graph, together with a sharper bound when the degree is sufficiently large. The authors suggest that the same cube-root phenomenon should hold for arbitrary graphs, but their stochastic analysis relies on regularity and does not establish this extension.

The proposed problem concerns constructing an online orientation algorithm for independently sampled edges from a non-regular graph and proving a discrepancy bound depending logarithmically on the number of arrivals, thereby extending the regular-graph stochastic result.

References

Theorem \ref{thm:uniform-rand-reg} suggests that, for $T$ edges sampled independently and uniformly from an arbitrary (possibly non-regular) graph $G$, there is an online algorithm with discrepancy $O(\log{1/3} T)$. We leave this as an interesting open question.

— The Cube-Root Phenomenon in Online Carpooling  (2609.21348 - Bansal et al., 18 Sep 2026) in Section 1, subsection “Our results,” paragraph labeled “Open question”