We show that the shortest $s$-$t$ path problem has the overlap-gap property in (i) sparse $\mathbf{G}(n,p)$ graphs and (ii) complete graphs with i.i.d. Exponential edge weights. Furthermore, we demonstrate that in sparse $\mathbf{G}(n,p)$ graphs, shortest path is solved by $O(\log n)$-degree polynomial estimators, and a uniform approximate shortest path can be sampled in polynomial time. This constitutes the first example in which the overlap-gap property is not predictive of algorithmic intractability for a (non-algebraic) average-case optimization problem.

翻译：我们证明了最短$s$-$t$路径问题在以下两类图中具有重叠间隙性质：(i)稀疏$\mathbf{G}(n,p)$随机图；(ii)具有独立同分布指数边权重的完全图。进一步地，我们证明在稀疏$\mathbf{G}(n,p)$图中，最短路径可由$O(\log n)$阶多项式估计器求解，且可在多项式时间内采样得到一致近似最短路径。这首次表明，对于（非代数）平均情形优化问题，重叠间隙性质并不能预测其算法难解性。