The inverse scattering problem for metric graphs and the traveling salesman problem

TL;DR

This paper solves the inverse scattering problem for differential Laplace operators on metric graphs, showing that the scattering matrix uniquely determines the graph's structure and boundary conditions, and applies this to the Traveling Salesman Problem.

Contribution

It introduces a novel combinatorial Fourier expansion of the scattering matrix to recover graph topology and boundary conditions, and applies this to combinatorial optimization problems.

Findings

Scattering matrix uniquely determines the graph and boundary conditions for almost all cases.

The approach encodes graph topology into the analytic properties of the scattering matrix.

An analytic method for solving the Traveling Salesman Problem on graphs is proposed.

Abstract

We present a solution to the inverse scattering problem for differential Laplace operators on metric noncompact graphs. We prove that for almost all boundary conditions (i) the scattering matrix uniquely determines the graph and its metric structure, (ii) the boundary conditions are determined uniquely up to trivial gauge transformations. The main ingredient of our approach is a combinatorial Fourier expansion of the scattering matrix which encodes the topology of the graph into analytic properties of the scattering matrix. Using the technique developed in this work, we also propose an analytic approach to solving some combinatorial problems on graphs, in particular, the Traveling Salesman Problem.