Sharp Logarithmic Thresholds for Cut Schedules in an Abstract Branch-and-Cut Model

Branch-and-cut interleaves branching with cutting-plane generation. How the two operations share the work of proving a bound is a basic theoretical question. We study an abstract model in which a tree certifies a target bound $Z$. Each branch node improves the bound by $\ell$ on one child and by $r$ on the other, where $0<\ell\le r$. The $i$th cut along a root-to-node path improves it by $c_i\ge0$, with cumulative improvement $C_k=\sum_{i=1}^k c_i$. Asymmetric branching enters through the rate $\lambda^{\star}>0$ defined by $e^{-\lambda^{\star}\ell}+e^{-\lambda^{\star}r}=1$. We establish uniform two-sided bounds of order $e^{\lambda^{\star}Z}$ on the minimal leaf count of pure branching trees. We then identify $\log k$ as the sharp threshold scale for the power of cutting. For cut schedules with extended limit $\gamma=\lim_{k\to\infty}C_k/\log k\in[0,\infty]$, minimal-size trees obey a trichotomy. If $\gamma=\infty$, cuts prove asymptotically all of the target. If $0\le\gamma<\infty$, the limiting fraction of the bound proved by cuts is $\gamma\lambda^{\star}/(1+\gamma\lambda^{\star})$. If $\gamma=0$, branch-and-cut has the same exponential size rate as pure branch-and-bound. This resolves open questions raised by Kazachkov, Le Bodic, and Sankaranarayanan on minimal-size trees under harmonically-worsening cuts, and generalizes their results to asymmetric branching and to all cut schedules in the model with this logarithmic limit. Finally, we show that branch-and-cut attains polynomial size in terms of $Z$ if and only if polynomially many cuts reduce the residual bound to $O(\log Z)$.