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A lasso-alternative to Dijkstra's algorithm for identifying short paths in networks

Anqi Dong, Amirhossein Taghvaei, Tryphon T. Georgiou

TL;DR

This work reframes the classic shortest-path problem as an $oldsymbol{\beta}$-regularized regression (lasso), enabling use of convex optimization techniques and solvers like LARS and ADMM. It reveals a deep link between the LARS solution path and bi-directional Dijkstra, showing that edge activation in the LARS path naturally builds two shortest-path trees that meet to form the optimal route. The paper further develops scalable proximal algorithms (ADMM and InADMM) suitable for very large graphs, and demonstrates through image, road-network, and synthetic geometric graphs that the approach can closely approximate shortest paths with favorable convergence and parallelization properties. Overall, it provides a unified optimization viewpoint on pathfinding with practical implications for dynamic graphs and distributed computation.

Abstract

We revisit the problem of finding the shortest path between two selected vertices of a graph and formulate this as an $\ell_1$-regularized regression -- Least Absolute Shrinkage and Selection Operator (lasso). We draw connections between a numerical implementation of this lasso-formulation, using the so-called LARS algorithm, and a more established algorithm known as the bi-directional Dijkstra. Appealing features of our formulation include the applicability of the Alternating Direction of Multiplier Method (ADMM) to the problem to identify short paths, and a relatively efficient update to topological changes.

A lasso-alternative to Dijkstra's algorithm for identifying short paths in networks

TL;DR

This work reframes the classic shortest-path problem as an -regularized regression (lasso), enabling use of convex optimization techniques and solvers like LARS and ADMM. It reveals a deep link between the LARS solution path and bi-directional Dijkstra, showing that edge activation in the LARS path naturally builds two shortest-path trees that meet to form the optimal route. The paper further develops scalable proximal algorithms (ADMM and InADMM) suitable for very large graphs, and demonstrates through image, road-network, and synthetic geometric graphs that the approach can closely approximate shortest paths with favorable convergence and parallelization properties. Overall, it provides a unified optimization viewpoint on pathfinding with practical implications for dynamic graphs and distributed computation.

Abstract

We revisit the problem of finding the shortest path between two selected vertices of a graph and formulate this as an -regularized regression -- Least Absolute Shrinkage and Selection Operator (lasso). We draw connections between a numerical implementation of this lasso-formulation, using the so-called LARS algorithm, and a more established algorithm known as the bi-directional Dijkstra. Appealing features of our formulation include the applicability of the Alternating Direction of Multiplier Method (ADMM) to the problem to identify short paths, and a relatively efficient update to topological changes.

Paper Structure

This paper contains 24 sections, 8 theorems, 60 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theoretic notations and definitions
  4. Shortest path problem and Dijkstra's algorithm
  5. Linear programming formulation
  6. Lasso formulation
  7. Uniqueness of the lasso solution
  8. The LARS algorithm
  9. Karush-Kuhn-Tucker (KKT) conditions
  10. The LARS algorithm
  11. Numerical example
  12. Relationship between lasso and Dijkstra
  13. Joining and crossing times
  14. Main result
  15. Proximal algorithm for large-scale graph
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\mathcal{G}$ be a tree rooted at $v_1$ with $n$ vertices and $P$ its path matrix. The pseudoinverse of its incidence matrix $D$, denoted as $D^{+}$, is Throughout, $\mathbbm{1}_k$ denotes the $k$-column vector with entries equal to $1$, and $I_k$ the $k\times k$ identity matrix. where $J=(I_{n-1}-\frac{1}{n} \mathbbm{1}_{n-1} \mathbbm{1}_{n-1}^T)$.

Figures (7)

  • Figure 1: The LARS algorithm successively identifies a set of active edges while reducing the tuning/control parameter $\lambda$. A vector $\beta(\lambda)$ with information of the length-contribution of the active edge-set is also successively being updated.
  • Figure 2: Path $\mathcal{P}_{s,t}$
  • Figure 3: "Edge detection" in an image as a short path, highlighted in red, and obtained using Dijkstra’s algorithm (Fig. \ref{['fig:Dijkstrapath']}), as well as using the lasso solution $\beta$ in ADMM and InADMM in Fig. \ref{['fig:ADMMpath']} and Fig. \ref{['fig:InADMMpath']}, respectively.
  • Figure 4: Example based on the image ($n = 4422$ and $m = 17291$): "Edge in image" identified via InADMM, InADMM with initializer, ADMM, and Basis pursuit. These converge in $36$, $34$, $29$, $47$ steps, respectively, with running times $1.7308$, $1.7618$, $4.0249$, $35.7527$ seconds in MATLAB clock. Also, the running times for solving the linear program \ref{['eq:linprog1']} using three methods ( dual-simplex and interior-point(-legacy)) are $26.6145$, $29.3866$, $32.2845$ seconds.
  • Figure 5: "Minimum-time trip" planning on Athens and Amsterdam road networks. Shortest paths from Dijkstra (yellow) and supports of the lasso solution $\beta$ recovered by ADMM (red) and InADMM (green) are overlaid.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2: Uniqueness
  • Remark 1: Consequences under Assumption \ref{['assum:st-uni']}
  • Proposition 3: Joining time
  • Proposition 4: Crossing time
  • Lemma 5: Edge adding
  • proof
  • Remark 2: No cycles
  • Lemma 6: No edge removed
  • ...and 6 more