Partially Disjoint k Shortest Paths

TL;DR

This paper explores the problem of finding multiple shortest paths in a graph with limited shared edges, focusing on near-shortest and exactly shortest paths, and extends results to multi-criteria weighted graphs.

Contribution

It introduces a new parameterized approach to the k shortest paths problem, considering the number of non-shared edges among paths, and generalizes to multi-criteria scenarios.

Findings

Analyzed paths with limited shared edges.

Extended results to multi-criteria weighted graphs.

Provided algorithms for near-shortest and exact shortest paths.

Abstract

A solution of the $k$ shortest paths problem may output paths that are identical up to a single edge. On the other hand, a solution of the $k$ independent shortest paths problem consists of paths that share neither an edge nor an intermediate node. We investigate the case in which the number of edges that are not shared among any two paths in the output $k$-set is a parameter. We study two main directions: exploring \emph{near-shortest} paths and exploring \emph{exactly shortest paths}. We assume that the weighted graph $G=(V,E,w)$ has no parallel edges and that the edge lengths (weights) are positive. Our results are also generalized to the cases of $k$ shortest paths where there are several weights per edge, and the results should take into account the multi-criteria prioritized weight.