Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Labeling Methods for Partially Ordered Paths (2307.10332v3)

Published 19 Jul 2023 in cs.DS and math.OC

Abstract: The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions' properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems' weight structure. Examples range from time-dependent shortest path and bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (41)
  1. Enhanced methods for the weight constrained shortest path problem: Constrained path finding meets bi-objective search. arXiv. https://doi.org/md43
  2. Shortest feasible paths with charging stops for battery electric vehicles. Transportation Science, 53(6), 1627–1655. https://doi.org/md4z
  3. Bellman, R. (1958). On a routing problem. Quarterly of Applied Mathematics, 16(1), 87–90. https://doi.org/ggb6qn
  4. Bertsekas, D. (2012). Dynamic programming and optimal control: Volume I. Athena Scientific.
  5. Bökler, F. K. (2018). Output-sensitive Complexity of Multiobjective Combinatorial Optimization with an Application to the Multiobjective Shortest Path Problem. TU Dortmund.
  6. Generalized dynamic programming for multicriteria optimization. European Journal of Operational Research, 44(1), 95–104. https://doi.org/dv2jb7
  7. The shortest route through a network with time-dependent internodal transit times. Journal of Mathematical Analysis and Applications, 14(3), 493–498. https://doi.org/czsr3j
  8. Shortest paths in networks with vector weights. Journal of Optimization Theory and Applications, 46, 79–86. https://doi.org/dzw768
  9. Lattices and complete lattices, 33–64. Cambridge University Press, (2 ed.). https://doi.org/md45
  10. Dean, B. C. (2004). Shortest paths in FIFO time-dependent networks: Theory and algorithms. Technical report, Massachusetts Institute of Technology.
  11. Round-based public transit routing. Transportation Science, 49(3), 591–604. https://doi.org/f7npcg
  12. Ehrgott, M. (2005). Multicriteria Optimization (2 ed.). Springer Berlin, Heidelberg. https://doi.org/cgtkmr
  13. Partial order approach to compute shortest paths in multimodal networks. arXiv. https://doi.org/md44
  14. Price optimal routing in public transportation. EURO Journal on Transportation and Logistics, 13, 100128. https://doi.org/10/mg88
  15. Fink, E. (1992). A survey of sequential and systolic algorithms for the algebraic path problem. Carnegie Mellon University.
  16. On the complexity of time-dependent shortest paths. Algorithmica, 68(4), 1075–1097. https://doi.org/f5sz42
  17. Martins’ algorithm revisited for multi-objective shortest path problems with a maxmin cost function. 4OR, 4(1), 47–59. https://doi.org/dn47sk
  18. The orienteering problem. Naval Research Logistics (NRL), 34(3), 307–318. https://doi.org/cn7fjd
  19. Hansen, P. (1980). Bicriterion path problems. Lecture Notes in Economics and Mathematical Systems, 177. https://doi.org/md47
  20. Simple and efficient bi-objective search algorithms via fast dominance checks. Artificial Intelligence, 314, 103807. https://doi.org/md5h
  21. Johnson, D. B. (1977). Efficient algorithms for shortest paths in sparse networks. Journal of the ACM, 24, 1–13. https://doi.org/dnvm2s
  22. Kahn, A. B. (1962). Topological sorting of large networks. Commun. ACM, 5(11), 558–562. https://doi.org/ch94wt
  23. Non-linear charge functions for electric vehicle scheduling with dynamic recharge rates (short paper). https://doi.org/md5g
  24. An FPTAS for dynamic multiobjective shortest path problems. Algorithms, 14(2), 1 – 22. https://doi.org/md49
  25. An improved multiobjective shortest path algorithm. Computers & Operations Research, 135, 105424. https://doi.org/md5j
  26. Martins, E. Q. V. (1984). On a multicriteria shortest path problem. European Journal of Operational Research, 16(2), 236–245. https://doi.org/bqsj2j
  27. Fuzzy shortest path problem. Computers & Industrial Engineering, 27(1), 465–468. https://doi.org/cz4tvz
  28. A shortest path problem on a network with fuzzy arc lengths. Fuzzy Sets and Systems, 109(1), 129–140. https://doi.org/cc66m3
  29. Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length. Journal of the ACM, 37(3), 607–625. https://doi.org/cwcbww
  30. Labeling methods for the general case of the multi-objective shortest path problem – a computational study. Computational Intelligence and Decision Making, 489–502.
  31. Parmentier, A. (2019). Algorithms for non-linear and stochastic resource constrained shortest path. Mathematical Methods of Operations Research, 89(2), 281–317. https://doi.org/md48
  32. A preference-based approach to spanning trees and shortest paths problems. European Journal of Operational Research, 162, 584–601. https://doi.org/bc3zx5
  33. Multi-objective and multi-constrained non-additive shortest path problems. Computers & Operations Research, 38(3), 605–616. https://doi.org/cc89g7
  34. A biobjective dijkstra algorithm. European Journal of Operational Research, 276(1), 106–118. https://doi.org/md5f
  35. Solving shortest path problems with a weight constraint and replenishment arcs. Computers & Operations Research, 39(5), 964–984. https://doi.org/b74bqw
  36. Tourist trip planning functionalities: State–of–the–art and future. Current Trends in Web Engineering, 474–485. https://doi.org/d2rxg6
  37. Storandt, S. (2012). Quick and energy-efficient routes: computing constrained shortest paths for electric vehicles. Proceedings of the 5th ACM SIGSPATIAL International Workshop on Computational Transportation Science, SIGSPATIAL’12. https://doi.org/md42
  38. Energy-efficient shortest routes for electric and hybrid vehicles. Transportation Research Part B: Methodological, 103, 111–135. https://doi.org/gbw7vp
  39. Szpilrajn, E. (1930). Sur l’extension de l’ordre partiel. Fundamenta Mathematicae, 16(1), 386–389. http://eudml.org/doc/212499
  40. On modelling and solving the shortest path problem with evidential weights. Belief Functions: Theory and Applications, 139–149. https://doi.org/md5d
  41. Anytime approximate bi-objective search. Proceedings of the International Symposium on Combinatorial Search, 15(1), 199–207. https://doi.org/md5b
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now