# On the Universal Near Shortest Simple Paths Problem

## TL;DR

This paper generalizes the Near Shortest Paths Problem using a universal weight vector, enabling enumeration of all universal near shortest simple paths and analyzing their complexity and cardinality.

## Contribution

It introduces a universal framework for near shortest paths, providing recursive algorithms for enumeration and complexity analysis.

## Key findings

- Algorithms efficiently enumerate universal near shortest simple paths.
- Complexity varies with different universal weight vectors.
- The cardinality of minimal complete sets is analyzed.

## Abstract

This article generalizes the Near Shortest Paths Problem introduced by Byers and Waterman in 1984 using concepts of the Universal Shortest Path Problem established by Turner and Hamacher in 2011. The generalization covers a variety of shortest path problems by introducing a universal weight vector. We apply this concept to the Near Shortest Paths Problem in a way that we are able to enumerate all universal near shortest simple paths. We present two recursive algorithms to compute the set of universal near shortest simple paths between two prespecified vertices and evaluate the running time complexity per path enumerated with respect to different values of the universal weight vector. Further, we study the cardinality of a minimal complete set with respect to different values of the universal weight vector.