On Constrained and k Shortest Paths

TL;DR

This paper examines the computational complexity of constrained and k shortest paths problems, highlighting their NP-hardness and providing ILP formulations, with implications for applications like data science dimensionality reduction.

Contribution

It analyzes two NP-hard shortest path variants and offers ILP formulations, providing new perspectives on solving these complex problems.

Findings

Both problems are NP-hard.

ILP formulations are proposed for these problems.

Relevance to data science applications like Isomap.

Abstract

Finding a shortest path in a graph is one of the most classic problems in algorithmic and graph theory. While we dispose of quite efficient algorithms for this ordinary problem (like the Dijkstra or Bellman-Ford algorithms), some slight variations in the problem statement can quickly lead to computationally hard problems. This article focuses specifically on two of these variants, namely the constrained shortest paths problem and the k shortest paths problem. Both problems are NP-hard, and thus it's not sure we can conceive a polynomial time algorithm (unless P = NP), ours aren't for instance. Moreover, across this article, we provide ILP formulations of these problems in order to give a different point of view to the interested reader. Although we did not try to implement these on modern ILP solvers, it can be an interesting path to explore. We also mention how these algorithms constitute essential ingredients in some of the most important modern applications in the field of data science, such as Isomap, whose main objective is the reduction of dimensionality of high-dimensional datasets.