LaplacianOpt Documentation

Overview

LaplacianOpt is a Julia package that implements polyhedral relaxation-based algorithms for the maximum algebraic connectivity augmentation problem on weighted graph Laplacians. More specifically, given a weighted base graph with existing edges (which could be empty), a set of candidate weighted edges for augmentation, and an augmentation budget (K), this package finds a set of K edges to augment the base graph such that the resulting graph has maximum algebraic connectivity with optimality guarantees. For example, given a base graph with N vertices and no edges, a set of candidate edges that forms a complete graph, and K = N - 1, this package finds a spanning tree with maximum algebraic connectivity.

Algebraic connectivity is the second smallest eigenvalue of the graph Laplacian. The magnitude of this value reflects how well connected the overall graph is. This connectivity measure has been used in analyzing the robustness and synchronizability of complex networks, and in graph sparsification techniques.

Installation

To use LaplacianOpt, first download and install Julia. The current version of LaplacianOpt requires Julia 1.10 or later.

The latest stable release of LaplacianOpt can be installed using the Julia package manager with

import Pkg
Pkg.add("LaplacianOpt")

At least one mixed-integer programming solver is required to run LaplacianOpt. CPLEX and Gurobi are recommended for solving larger graph instances. The open-source GLPK solver is also compatible with LaplacianOpt and can be installed through the package manager with

import Pkg
Pkg.add("GLPK")

Unit Tests

To run the tests in the package, run the following command after installing the LaplacianOpt package.

import Pkg
Pkg.test("LaplacianOpt")

Citing LaplacianOpt

If you find LaplacianOpt useful in your work, please cite the publication(s) most relevant to your use of the package. The Research & Publications page provides a chronological overview of the research, paper links, and copyable BibTeX citations.