Greedy cycle upper bound
Establish that greedy allocation on the cycle graph C_n, started from the flat load profile, has uniformly bounded expected gap of order O(\sqrt{n}) over all times, and determine whether every invariant law of the normalized process satisfies the corresponding stationary O(\sqrt{n}) bound.
References
The lower bound on the cycle has the expected saturated scale, but the corresponding upper bound for greedy graphical allocation remains open. The best known equilibrium upper bound is $O(n)$ , so the stationary gap is presently known only between order $\sqrt n$ and order $n$. \begin{problem}[Greedy cycle upper bound] Does greedy allocation on $C_n$, started from the flat profile, satisfy
\sup_{t\ge0}EGap(X_t)=O(\sqrt n)?
Does every invariant law of the normalized process satisfy the corresponding stationary bound? A stationary estimate alone does not imply the uniform transient estimate without an additional convergence argument.\end{problem}
Finally, a rule that reads loads in a bounded neighborhood need not preserve one unit discrepancy: an off-edge perturbation may change a remote endpoint decision and create a signed discrepancy with several nonzero coordinates. The obstruction is therefore not captured by a single tag.
\begin{problem}[Bounded-radius information] Find a replacement for the unit-discrepancy tag for rules whose decision on an edge may inspect loads within graph distance $r_0$, where $r_0$ is fixed. Does every such monotone rule still incur a polynomial gap on the cycle, and what exponent is forced by the propagation of its signed response? \end{problem}