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Improved Approximation Algorithms for Bounded-Length Path Augmentation

Author Felix Hommelsheim

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LIPIcs.APPROX-RANDOM.2026.7.pdf
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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Routing and network design problems
Keywords
  • Approximation algorithms
  • path augmentation
  • connectivity augmentation
  • survivable network design

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Abstract

The Forest Augmentation Problem (FAP) asks for a minimum set of additional edges (links) that makes a given forest 2-edge-connected while spanning all vertices. A key special case is the Path Augmentation Problem (PAP), where the input forest consists of vertex-disjoint paths. Grandoni, Jabal Ameli, and Traub [STOC'22] recently broke the long-standing 2-approximation barrier for FAP, achieving a 1.9973-approximation. A crucial component of this result was their 1.9913-approximation for PAP; the first better-than-2 approximation for PAP.
In this work, we present a 1.9412-approximation for bounded-length PAP, in which each path of the input forest has constant length p_max. The running-time of our algorithm is O(n^p_max). One of our key innovations is a (11/6 + ε)-approximation preserving reduction to so-called structured instances, which simplifies the problem and enables our improved approximation. Additionally, we introduce a new relaxation inspired by 2-edge covers and analyze it via a corresponding packing problem, where the relationship between the two problems is similar to the relationship between 2-edge covers and 2-matchings. Using a factor-revealing LP, we bound the cost of our solution to the packing problem w.r.t. the relaxation and derive a strong initial solution. We then transform this solution into a feasible PAP solution, combining techniques from FAP and related connectivity augmentation problems, along with new insights.

Cite As Get BibTex

Felix Hommelsheim. Improved Approximation Algorithms for Bounded-Length Path Augmentation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.7

Author Details

Felix Hommelsheim
  • University of Cologne, Germany

Funding

The work of the first author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) by the grant HO 7562/2-1 (project ID: 573939419).

References

  1. David Adjiashvili. Beating approximation factor two for weighted tree augmentation with bounded costs. ACM Transactions on Algorithms (TALG), 15(2):1-26, 2018. URL: https://doi.org/10.1145/3182395.
  2. Étienne Bamas, Marina Drygala, and Ola Svensson. A simple lp-based approximation algorithm for the matching augmentation problem. In Karen I. Aardal and Laura Sanità, editors, Integer Programming and Combinatorial Optimization - 23rd International Conference, IPCO 2022, Eindhoven, The Netherlands, June 27-29, 2022, Proceedings, volume 13265 of Lecture Notes in Computer Science, pages 57-69. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06901-7_5.
  3. Miguel Bosch-Calvo, Mohit Garg, Fabrizio Grandoni, Felix Hommelsheim, Afrouz Jabal Ameli, and Alexander Lindermayr. A 5/4-approximation for two-edge connectivity. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 653-664, 2025. URL: https://doi.org/10.1145/3717823.3718275.
  4. Miguel Bosch-Calvo, Fabrizio Grandoni, and Afrouz Jabal Ameli. A 4/3 approximation for 2-vertex-connectivity. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 29:1-29:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.29.
  5. Jaroslaw Byrka, Fabrizio Grandoni, and Afrouz Jabal Ameli. Breaching the 2-approximation barrier for connectivity augmentation: A reduction to steiner tree. SIAM J. Comput., 52(3):718-739, 2023. URL: https://doi.org/10.1137/21M1421143.
  6. Federica Cecchetto, Vera Traub, and Rico Zenklusen. Bridging the gap between tree and connectivity augmentation: unified and stronger approaches. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 370-383, 2021. URL: https://doi.org/10.1145/3406325.3451086.
  7. Joe Cheriyan, Jack Dippel, Fabrizio Grandoni, Arindam Khan, and Vishnu V Narayan. The matching augmentation problem: a 7 4-approximation algorithm. Mathematical Programming, 182(1-2):315-354, 2020. URL: https://doi.org/10.1007/s10107-019-01394-z.
  8. Joseph Cheriyan, Robert Cummings, Jack Dippel, and Jasper Zhu. An improved approximation algorithm for the matching augmentation problem. SIAM Journal on Discrete Mathematics, 37(1):163-190, 2023. URL: https://doi.org/10.1137/21M1453505.
  9. Joseph Cheriyan and Zhihan Gao. Approximating (unweighted) tree augmentation via lift-and-project, part i: stemless tap. Algorithmica, 80:530-559, 2018. URL: https://doi.org/10.1007/s00453-016-0270-4.
  10. Joseph Cheriyan and Zhihan Gao. Approximating (unweighted) tree augmentation via lift-and-project, part ii. Algorithmica, 80:608-651, 2018. URL: https://doi.org/10.1007/s00453-017-0275-7.
  11. Joseph Cheriyan, Howard Karloff, Rohit Khandekar, and Jochen Könemann. On the integrality ratio for tree augmentation. Operations Research Letters, 36(4):399-401, 2008. URL: https://doi.org/10.1016/j.orl.2008.01.009.
  12. Joseph Cheriyan, András Sebo, and Zoltán Szigeti. Improving on the 1.5-approximation of a smallest 2-edge connected spanning subgraph. SIAM Journal on Discrete Mathematics, 14(2):170-180, 2001. URL: https://doi.org/10.1137/S0895480199362071.
  13. Joseph Cheriyan and Ramakrishna Thurimella. Approximating minimum-size k-connected spanning subgraphs via matching. SIAM J. Comput., 30(2):528-560, 2000. URL: https://doi.org/10.1137/S009753979833920X.
  14. Nachshon Cohen and Zeev Nutov. A (1+ ln2)-approximation algorithm for minimum-cost 2-edge-connectivity augmentation of trees with constant radius. Theoretical Computer Science, 489:67-74, 2013. URL: https://doi.org/10.1016/j.tcs.2013.04.004.
  15. Marek Cygan. Improved approximation for 3-dimensional matching via bounded pathwidth local search. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 509-518. IEEE, 2013. URL: https://doi.org/10.1109/FOCS.2013.61.
  16. Artur Czumaj and Andrzej Lingas. On approximability of the minimum-cost k-connected spanning subgraph problem. In Robert Endre Tarjan and Tandy J. Warnow, editors, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 17-19 January 1999, Baltimore, Maryland, USA, pages 281-290. ACM/SIAM, 1999. URL: https://doi.org/10.5555/314500.314573.
  17. Guy Even, Jon Feldman, Guy Kortsarz, and Zeev Nutov. A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2. ACM Transactions on Algorithms (TALG), 5(2):1-17, 2009. URL: https://doi.org/10.1145/1497290.1497297.
  18. Cristina G. Fernandes. A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem. J. Algorithms, 28(1):105-124, 1998. URL: https://doi.org/10.1006/jagm.1998.0931.
  19. Samuel Fiorini, Martin Groß, Jochen Könemann, and Laura Sanità. Approximating weighted tree augmentation via chvátal-gomory cuts. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 817-831. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.53.
  20. Greg N Frederickson and Joseph Ja’Ja’. Approximation algorithms for several graph augmentation problems. SIAM Journal on Computing, 10(2):270-283, 1981. URL: https://doi.org/10.1137/0210019.
  21. Martin Fürer and Huiwen Yu. Approximating the-set packing problem by local improvements. In International Symposium on Combinatorial Optimization, pages 408-420. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-09174-7_35.
  22. Mohit Garg, Fabrizio Grandoni, and Afrouz Jabal Ameli. Improved approximation for two-edge-connectivity. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2368-2410. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch92.
  23. Mohit Garg, Felix Hommelsheim, and Nicole Megow. Matching augmentation via simultaneous contractions. In 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, pages 65:1-65:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.65.
  24. Naveen Garg, Santosh S. Vempala, and Aman Singla. Improved approximation algorithms for biconnected subgraphs via better lower bounding techniques. In Vijaya Ramachandran, editor, Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 25-27 January 1993, Austin, Texas, USA, pages 103-111. ACM/SIAM, 1993. URL: https://doi.org/10.5555/313559.313618.
  25. Fabrizio Grandoni, Afrouz Jabal Ameli, and Vera Traub. Breaching the 2-approximation barrier for the forest augmentation problem. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1598-1611, 2022. URL: https://doi.org/10.1145/3519935.3520035.
  26. Fabrizio Grandoni, Christos Kalaitzis, and Rico Zenklusen. Improved approximation for tree augmentation: saving by rewiring. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 632-645, 2018. URL: https://doi.org/10.1145/3188745.3188898.
  27. Klaus Heeger and Jens Vygen. Two-connected spanning subgraphs with at most 10/7OPT edges. SIAM J. Discret. Math., 31(3):1820-1835, 2017. URL: https://doi.org/10.1137/16M1091587.
  28. Felix Hommelsheim. Improved approximation algorithms for path and forest augmentation via a novel relaxation. CoRR, abs/2505.15324, 2025. URL: https://doi.org/10.48550/arXiv.2505.15324.
  29. Felix Hommelsheim, Alexander Lindermayr, and Zhenwei Liu. A better-than-5/4-approximation for two-edge connectivity. In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1468-1520. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.54.
  30. Christoph Hunkenschröder, Santosh Vempala, and Adrian Vetta. A 4/3-approximation algorithm for the minimum 2-edge connected subgraph problem. ACM Transactions on Algorithms (TALG), 15(4):1-28, 2019. URL: https://doi.org/10.1145/3341599.
  31. Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21:39-60, 2001. URL: https://doi.org/10.1007/s004930170004.
  32. Samir Khuller and Ramakrishna Thurimella. Approximation algorithms for graph augmentation. Journal of algorithms, 14(2):214-225, 1993. URL: https://doi.org/10.1006/jagm.1993.1010.
  33. Samir Khuller and Uzi Vishkin. Biconnectivity approximations and graph carvings. Journal of the ACM (JACM), 41(2):214-235, 1994. URL: https://doi.org/10.1145/174652.174654.
  34. David G Kirkpatrick and Pavol Hell. On the completeness of a generalized matching problem. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages 240-245, 1978. URL: https://doi.org/10.1145/800133.804353.
  35. Yusuke Kobayashi and Takashi Noguchi. An approximation algorithm for two-edge-connected subgraph problem via triangle-free two-edge-cover. arXiv preprint arXiv:2304.13228, 2023. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2023.49.
  36. Guy Kortsarz and Zeev Nutov. A simplified 1.5-approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2. ACM Transactions on Algorithms (TALG), 12(2):1-20, 2015. URL: https://doi.org/10.1145/2786981.
  37. Guy Kortsarz and Zeev Nutov. Lp-relaxations for tree augmentation. Discrete Applied Mathematics, 239:94-105, 2018. URL: https://doi.org/10.1016/j.dam.2017.12.033.
  38. Hiroshi Nagamochi. An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree. Discrete Applied Mathematics, 126(1):83-113, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00218-4.
  39. Zeev Nutov. On the tree augmentation problem. Algorithmica, 83:553-575, 2021. URL: https://doi.org/10.1007/s00453-020-00765-9.
  40. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer, Berlin, Heidelberg, 2003. URL: https://link.springer.com/book/9783540443896.
  41. András Sebő and Jens Vygen. Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica, pages 1-34, 2014. URL: https://doi.org/10.1007/s00493-014-2960-3.
  42. Vera Traub and Rico Zenklusen. Better-than-2 approximations for weighted tree augmentation and applications to steiner tree. J. ACM, 72(2):16:1-16:40, 2025. URL: https://doi.org/10.1145/3722101.
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