Lasso formulation of the shortest path problem

TL;DR

This paper introduces a novel lasso-based formulation of the shortest path problem, establishing a connection with Dijkstra's algorithm and proposing an ADMM-based method for efficient approximation in large graphs.

Contribution

It formulates shortest path as an l_1-regularized regression problem, linking it to existing algorithms and proposing a scalable approximation method using ADMM.

Findings

The lasso formulation aligns with Dijkstra's shortest path trees.

ADMM provides fast approximate solutions for large graphs.

Numerical experiments demonstrate the method's effectiveness.

Abstract

The shortest path problem is formulated as an $l_1$-regularized regression problem, known as lasso. Based on this formulation, a connection is established between Dijkstra's shortest path algorithm and the least angle regression (LARS) for the lasso problem. Specifically, the solution path of the lasso problem, obtained by varying the regularization parameter from infinity to zero (the regularization path), corresponds to shortest path trees that appear in the bi-directional Dijkstra algorithm. Although Dijkstra's algorithm and the LARS formulation provide exact solutions, they become impractical when the size of the graph is exceedingly large. To overcome this issue, the alternating direction method of multipliers (ADMM) is proposed to solve the lasso formulation. The resulting algorithm produces good and fast approximations of the shortest path by sacrificing exactness that may not be absolutely essential in many applications. Numerical experiments are provided to illustrate the performance of the proposed approach.