A rigorous analysis of the cavity equations for the minimum spanning tree

TL;DR

This paper introduces a new local-constraint-based representation for the MST problem, demonstrating that cavity equations can reliably find the global optimum in the spanning tree case.

Contribution

It develops a novel local conditions framework for MST and proves cavity equations' fixed points yield the optimal solution for spanning trees.

Findings

Cavity equations fixed points correspond to the global optimum for spanning trees.

New local representation simplifies analysis of MST constraints.

The approach potentially extends to other combinatorial optimization problems.

Abstract

We analyze a new general representation for the Minimum Weight Steiner Tree (MST) problem which translates the topological connectivity constraint into a set of local conditions which can be analyzed by the so called cavity equations techniques. For the limit case of the Spanning tree we prove that the fixed point of the algorithm arising from the cavity equations leads to the global optimum.