Some easy optimization problems have the overlap-gap property

TL;DR

This paper investigates the overlap-gap property in simple optimization problems like shortest path, showing it does not necessarily imply computational difficulty, and introduces efficient algorithms for certain graph models.

Contribution

It demonstrates that the overlap-gap property does not always indicate algorithmic intractability, providing new insights into average-case optimization problems.

Findings

Shortest path problem exhibits overlap-gap property in specific graph models.

Efficient polynomial estimators can solve shortest path in sparse graphs.

Sampling approximate shortest paths is feasible in polynomial time.

Abstract

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.