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

TL;DR

The paper investigates the average-case complexity of two natural optimization problems on random graphs: finding a shortest path between two nodes ($P_1$) and minimizing the sum of root-path distances over all nodes in a spanning tree ($P_2$). It develops two complementary frameworks—ensemble overlap-gap properties (OGP) and finite-temperature Franz–Parisi potential (FPP)—to predict the behavior of local search algorithms, and shows that while P1 exhibits an OGP barrier (yet remains polynomial-time solvable), P2 lacks an ensemble OGP and has no Franz–Parisi energy barrier. The analysis combines distance distribution, overlap asymptotics on correlated Erdős–Rényi graphs, and a precise Gibbs-measure/partition-function treatment that reduces to a one-dimensional variational problem, leading to explicit phase diagrams and replica-symmetric behavior away from criticality. The results illuminate how landscape structure informs algorithmic tractability and connect to submodular minimization via the Lovász extension, offering a unified view of stability, phase transitions, and metastability in random combinatorial optimization. These insights advance understanding of when local-search strategies fail or succeed in sparse random settings and provide a rigorous bridge between statistical physics heuristics and concrete combinatorial problems.

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.

Figures (13)

• Figure 1: Numerical simulation of the overlap of uniformly random shortest path trees vs. shortest paths. It was conducted independently three times with parameters $n=10^5$ and $q=10^{-4}$.
• Figure 2: Phase diagram in the limit $n\to\infty$. The second diagram is obtained by "zooming in" the first diagram at $1-\Delta=1$, and the third diagram is from zooming in the second diagram at $\lambda=1$. The labels for the regions correspond to the regimes \ref{['item:intro-regime1']} through \ref{['item:intro-regime3-3']}. The blue region is the low-temperature phase where the Gibbs measure is close to the uniform measure in Wasserstein distance, and the red region is the high-temperature phase where they are far apart. Solid black lines indicate a discontinuous ("first-order") phase transition, and dashed lines indicate a continuous ("second-order") phase transition. The point at which the line changes from solid to dashed is precisely where the parameter regime transitions from $\lambda<1$ to $\lambda=1$.
• Figure 3: Visualization of the potential well for different regimes. In the middle well, the high temperature state and the low temperature state are separated by a gray region with edges of constant slope $\beta_c^{-1}$, resulting in a discontinuous phase transition. For the rightmost one, the two states touch each other, leading to a continuous phase transition.
• Figure 4: A plot of the Franz--Parisi potential against the correlation $r\in[0,1]$ for different regimes. The $y$-axes (for $\mathcal{F}_\beta^{\text{FP}}$) are scaled differently to better visualize the landscape. For the right plot we choose $\lambda=0.3$ (and $\Delta=0$). A low temperature phase $\beta=1.5$ in the right plot has its unique quasiconvex landscape which is quite different from the other cases which simply increase in $r$.
• Figure 5: Uniformly random spanning tree overlap $\tilde{R}_n$ in its two regimes. Top: $\gamma > 0$, so $\rho = 1$. Bottom: $\gamma = 0$, so $\rho \in [0,1]$. The bottom of the top figure and top of the bottom figure coincide.

Theorems & Definitions (233)

• Definition 1.1: gamarnik2021overlapchen2019suboptimality
• Theorem 1.1: Distance asymptotics
• Theorem 1.2: Correlated graph asymptotics
• Corollary 1.3: No e-OGP for shortest path tree
• proof
• Corollary 1.4: Wasserstein Disorder Chaos
• proof
• Theorem 1.5: Disorder chaos
• Remark : Two phases of evolution
• Remark