The Shortest Path Problem with Edge Information Reuse is NP-Complete

TL;DR

This paper proves that a variant of the shortest path problem, involving edge information reuse, is NP-complete, by reducing from 3SAT, highlighting its computational complexity.

Contribution

The paper introduces and proves NP-completeness of the edge information reuse shortest path problem, a novel variation of the classical shortest path problem.

Findings

The problem is NP-complete via reduction from 3SAT.

Edge information reuse affects shortest path computation complexity.

The problem involves counting edge weights with information reuse constraints.

Abstract

We show that the following variation of the single-source shortest path problem is NP-complete. Let a weighted, directed, acyclic graph $G=(V,E,w)$ with source and sink vertices $s$ and $t$ be given. Let in addition a mapping $f$ on $E$ be given that associates information with the edges (e.g., a pointer), such that $f(e)=f(e')$ means that edges $e$ and $e'$ carry the same information; for such edges it is required that $w(e)=w(e')$. The length of a simple $st$ path $U$ is the sum of the weights of the edges on $U$ but edges with $f(e)=f(e')$ are counted only once. The problem is to determine a shortest such $st$ path. We call this problem the \emph{edge information reuse shortest path problem}. It is NP-complete by reduction from 3SAT.