Some easy optimization problems have the overlap-gap property

2024-11-04

arXiv.org

AI Summary

This work investigates whether the shortest $s$-$t$ path problem exhibits the Overlap-Gap Property (OGP) in sparse random graphs and in complete graphs with i.i.d. exponential edge weights, and how OGP relates to computational hardness. Using tools from random graph theory, polynomial estimator analysis, and uniform sampling algorithm design, we establish—contrary to expectation—that although OGP holds for this problem, it does not imply average-case computational intractability. Our key contributions are: (i) the construction of an $O(log n)$-degree polynomial estimator that exactly computes the shortest $s$-$t$ path, and (ii) a polynomial-time algorithm for uniform approximate sampling of shortest paths. This is the first demonstration, in a non-algebraic average-case optimization setting, that OGP does not entail algorithmic hardness—thereby challenging the prevailing view of OGP as a universal computational barrier.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Proving overlap-gap property in shortest path problems
• Analyzing sparse and complete graphs with Exponential weights
• Demonstrating polynomial-time solvability despite overlap-gap property

Innovation

• Overlap-gap property in sparse and complete graphs
• Solved by O(log n)-degree polynomial estimators
• Uniform approximate sampling in polynomial time