The minimum reload $s$–$t$ path, trail and walk problems

Abstract

This paper deals with problems on non-oriented edge-colored graphs. The aim is to find a route between two given vertices $s$ and $t$. This route can be a walk, a trail or a path. Each time a vertex is crossed by a walk there is an associated non-negative reload cost $r_{i,j}$, where $i$ and $j$ denote, respectively, the colors of successive edges in this walk. The goal is to find a route whose total reload cost is minimum. Polynomial algorithms and proofs of NP-hardness are given for particular cases: when the triangle inequality is satisfied or not, when reload costs are symmetric (i.e., $r_{i,j}=r_{j,i}$) or asymmetric. We also investigate bounded degree graphs and planar graphs. We conclude the paper with the traveling salesman problem with reload costs.