Algorithmica https://doi.org/10.1007/s00453-018-0414-9 Path Refinement in Weighted Regions Amin Gheibi 1,2 · Anil Maheshwari 1 · Jörg-Rüdiger Sack 1 · Christian Scheffer 3 Received: 13 August 2015 / Accepted: 6 February 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this paper, we study the weighted region problem (WRP) which is to compute a shortest path in a weighted partitioning of a plane. Recent results show that WRP is not solvable in any algebraic computation model over the rational numbers. Therefore, it is unlikely that WRP can be solved in polynomial time. Research has thus focused on determining approximate solutions for WRP. Approximate solutions for WRP typically show qualitatively different behaviors. We first formulate two qual- itative criteria for weighted shortest paths. Then, we show how to produce a path that is quantitatively close-to-optimal and qualitatively satisfactory. More precisely, we propose an algorithm to transform any given approximate linear path into a linear path with the same (or shorter) weighted length for which we can prove that it satisfies the required qualitative criteria. This algorithm has a linear time complexity in the size of the given path. At the end, we explain our experiments on some triangular irregular Research supported by Natural Sciences and Engineering Research Council of Canada. B Amin Gheibi agheibi@scs.carleton.ca Anil Maheshwari anil@scs.carleton.ca Jörg-Rüdiger Sack sack@scs.carleton.ca Christian Scheffer scheffer@ibr.cs.tu-bs.de 1 School of Computer Science, Carleton University, Ottawa, ON, Canada 2 Amirkabir University of Technology, Tehran, Iran 3 Institute of Operating Systems and Computer Networks, Technische Universität Braunschweig, Brunswick, Germany 123