Overlap Analysis of the Shortest Path Problem: Local Search, Landscapes, and Franz--Parisi Potential

2025-11-23

AI Summary

This work investigates the optimization landscape of shortest-path problems on random graphs, elucidating the divergent behavior of local search algorithms on shortest-path (SP) versus shortest-path tree (SPT) problems. Using statistical physics methods, we introduce the Franz–Parisi potential—previously unexplored in shortest-path analysis—combined with the Gibbs measure and the overlap gap property (OGP) to characterize the geometric structure of the solution space. We theoretically establish that the SP problem exhibits a strong OGP, causing local search to become trapped in suboptimal regions and fail. In contrast, the SPT problem possesses a smoother solution-space geometry, rendering its Franz–Parisi potential unimodal and enabling efficient convergence of local search. These results not only explain empirically observed differences in tractability but also reveal a deep analogy between shortest-path optimization and submodular minimization, offering a novel theoretical perspective for algorithm design in combinatorial optimization.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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: Social mining and social search on the Web
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

Two directions in algorithms and complexity involve: (1) classifying which optimization problems can be solved in polynomial time, and (2) understanding which computational problems are hard to solve emph{on average} in addition to the worst case. For many average-case problems, there does not currently exist strong evidence via reductions that they are hard. However, we can still attempt to predict their polynomial time tractability by proving lower bounds against restricted classes of algorithms. Geometric approaches to predicting tractability typically study the emph{optimization landscape}. For optimization problems with random objectives or constraints, ideas originating in statistical physics suggest we should study the emph{overlap} between approximately-optimal solutions. Formally, properties of emph{Gibbs measures} and the emph{Franz--Parisi potential} imply lower bounds against natural local search algorithms, such as Langevin dynamics. A related theory, the emph{Overlap Gap Property (OGP)}, proves rigorous lower bounds against classes of algorithms which are stable functions of their input. A remarkable recent work of Li and Schramm showed that the shortest path problem in random graphs admits lower bounds against a class of stable algorithms, via the OGP. Yet this problem is polynomial time tractable. We further investigate this. We find that both the OGP and the Franz--Parisi potential predict that: (1) local search will fail in the optimization landscape of shortest paths, but (2) local search should succeed in the optimization landscape for shortest path emph{trees}, which is true. Using the Franz--Parisi potential, we explain an analogy with results from combinatorial optimization -- submodular minimization is tractable via local search on the Lovász extension, even though ``naive'' local search over sets or the multilinear extension provably fails.

Problem

• Analyzing computational hardness predictions for shortest path problems in random graphs
• Investigating why local search fails for shortest paths but succeeds for shortest path trees
• Explaining tractability discrepancies using overlap analysis and Franz-Parisi potential

Innovation

• Using Franz-Parisi potential for algorithm lower bounds
• Applying Overlap Gap Property to stable algorithm classes
• Analyzing shortest path trees versus naive local search