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K-Hole Separation in PEO‑Based ILP Treewidth Formulation

Author Andrea D'Ascenzo

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Subject Classification

ACM Subject Classification
  • Theory of computation → Integer programming
  • Mathematics of computing → Graph theory
Keywords
  • Treewidth
  • Integer Linear Programming
  • Polyhedral Combinatorics
  • Chordal Completion
  • Induced Cycles

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Abstract

In this paper, we introduce a family of valid inequalities for the strongest currently known integer programming formulation of treewidth based on perfect elimination orderings. These inequalities arise from the structure of induced chordless cycles (holes) and strengthen the canonical linear relaxation by enforcing constraints that every feasible chordal completion must satisfy. To handle the exponentially many such inequalities, we develop a dedicated separation routine capable of detecting violated k-hole constraints within a cutting-plane framework. Our computational results show that incorporating these inequalities substantially improves the quality of the lower bounds across a broad range of graph classes, in some cases nearly closing the integrality gap.

Cite As Get BibTex

Andrea D'Ascenzo. K-Hole Separation in PEO‑Based ILP Treewidth Formulation. In 24th International Symposium on Experimental Algorithms (SEA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 371, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SEA.2026.14

Author Details

Andrea D'Ascenzo
  • Department of Computer Science, Gran Sasso Science Institute, L'Aquila, Italy

Funding

This work is partially supported by the MUR (Italy) Department of Excellence 2023–2027.

Acknowledgements

The author is grateful to the reviewers for their helpful comments.

Supplementary Materials

References

  1. Ahmad Turki Anaqreh, G Boglárka, Tamás Vinkó, et al. Exact methods for the longest induced cycle problem. Croatian Operational Research Review, 15(2):199-212, 2024. Google Scholar
  2. Max Bannach, Sebastian Berndt, and Thorsten Ehlers. Jdrasil: A modular library for computing tree decompositions. In Costas S. Iliopoulos, Solon P. Pissis, Simon J. Puglisi, and Rajeev Raman, editors, 16th International Symposium on Experimental Algorithms, SEA 2017, London, UK, June 21-23, 2017, volume 75 of LIPIcs, pages 28:1-28:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.SEA.2017.28.
  3. David Bergman, Carlos Henrique Cardonha, André Augusto Ciré, and Arvind U. Raghunathan. On the minimum chordal completion polytope. Oper. Res., 67(2):532-547, 2019. URL: https://doi.org/10.1287/OPRE.2018.1783.
  4. David Bergman and Arvind U. Raghunathan. A benders approach to the minimum chordal completion problem. In Laurent Michel, editor, Integration of AI and OR Techniques in Constraint Programming - 12th International Conference, CPAIOR 2015, Barcelona, Spain, May 18-22, 2015, Proceedings, volume 9075 of Lecture Notes in Computer Science, pages 47-64. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-18008-3_4.
  5. Andrew Beveridge and Jie Shan. Network of thrones. Math Horizons, 23(4):18-22, 2016. Google Scholar
  6. Hans L. Bodlaender. Dynamic programming on graphs with bounded treewidth. In Timo Lepistö and Arto Salomaa, editors, Automata, Languages and Programming, 15th International Colloquium, ICALP88, Tampere, Finland, July 11-15, 1988, Proceedings, volume 317 of Lecture Notes in Computer Science, pages 105-118. Springer, 1988. URL: https://doi.org/10.1007/3-540-19488-6_110.
  7. Hans L. Bodlaender. Discovering treewidth. In Peter Vojtás, Mária Bieliková, Bernadette Charron-Bost, and Ondrej Sýkora, editors, SOFSEM 2005: Theory and Practice of Computer Science, 31st Conference on Current Trends in Theory and Practice of Computer Science, Liptovský Ján, Slovakia, January 22-28, 2005, Proceedings, volume 3381 of Lecture Notes in Computer Science, pages 1-16. Springer, 2005. URL: https://doi.org/10.1007/978-3-540-30577-4_1.
  8. Hans L. Bodlaender and Arie M. C. A. Koster. Combinatorial optimization on graphs of bounded treewidth. Comput. J., 51(3):255-269, 2008. URL: https://doi.org/10.1093/COMJNL/BXM037.
  9. Claudson F. Bornstein and Santosh S. Vempala. Flow metrics. Theor. Comput. Sci., 321(1):13-24, 2004. URL: https://doi.org/10.1016/J.TCS.2003.05.003.
  10. David Coudert. A note on integer linear programming formulations for linear ordering problems on graphs, 2016. Google Scholar
  11. David Coudert, Andrea D'Ascenzo, and Clément Rambaud. k-shortest simple paths in bounded treewidth graphs. Theor. Comput. Sci., 1039:115182, 2025. URL: https://doi.org/10.1016/J.TCS.2025.115182.
  12. Holger Dell, Thore Husfeldt, Bart M. P. Jansen, Petteri Kaski, Christian Komusiewicz, and Frances A. Rosamond. The First Parameterized Algorithms and Computational Experiments Challenge. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016), volume 63 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1-30:9, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.30.
  13. Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration. In Daniel Lokshtanov and Naomi Nishimura, editors, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), volume 89 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1-30:12, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.30.
  14. Iain S. Duff, Roger G. Grimes, and John G. Lewis. Sparse matrix test problems. ACM Trans. Math. Softw., 15(1):1-14, 1989. URL: https://doi.org/10.1145/62038.62043.
  15. Jack Edmonds. Maximum matching and a polyhedron with 0, 1-vertices. Journal of research of the National Bureau of Standards B, 69(125-130):55-56, 1965. Google Scholar
  16. David Eppstein and Denis Kurz. K-best solutions of MSO problems on tree-decomposable graphs. In Daniel Lokshtanov and Naomi Nishimura, editors, 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, Vienna, Austria, September 6-8, 2017, volume 89 of LIPIcs, pages 16:1-16:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.16.
  17. Paul Erd6s and Alfréd Rényi. On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci, 5:17-61, 1960. Google Scholar
  18. Baris Can Esmer, Jacob Focke, Dániel Marx, and Pawel Rzazewski. Fundamental problems on bounded-treewidth graphs: The real source of hardness. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, Tallinn, Estonia, July 8-12, 2024, volume 297 of LIPIcs, pages 34:1-34:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.34.
  19. Linton C Freeman, Sue C Freeman, and Alaina G Michaelson. On human social intelligence. Journal of Social and Biological Structures, 11(4):415-425, 1988. Google Scholar
  20. Alexander Grigoriev. Possible and impossible attempts to solve the treewidth problem via ilps. In Fedor V. Fomin, Stefan Kratsch, and Erik Jan van Leeuwen, editors, Treewidth, Kernels, and Algorithms - Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday, volume 12160 of Lecture Notes in Computer Science, pages 78-88. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_7.
  21. Alexander Grigoriev, Hans Ensinck, and Natalya Usotskaya. Integer linear programming formulations for treewidth. Maastricht Research School of Economics of Technology and Organizations, 2011. Google Scholar
  22. Martin Grötschel, Michael Jünger, and Gerhard Reinelt. A cutting plane algorithm for the linear ordering problem. Oper. Res., 32(6):1195-1220, 1984. URL: https://doi.org/10.1287/OPRE.32.6.1195.
  23. Martin Grötschel, Michael Jünger, and Gerhard Reinelt. Facets of the linear ordering polytope. Math. Program., 33(1):43-60, 1985. URL: https://doi.org/10.1007/BF01582010.
  24. Martin Grötschel and George L. Nemhauser. A polynomial algorithm for the max-cut problem on graphs without long odd cycles. Math. Program., 29(1):28-40, 1984. URL: https://doi.org/10.1007/BF02591727.
  25. Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2023. URL: https://www.gurobi.com.
  26. Christine C Hass. Social status in female bighorn sheep (ovis canadensis): expression, development and reproductive correlates. Journal of Zoology, 225(3):509-523, 1991. Google Scholar
  27. Michael Jünger and Sven Mallach. Odd-cycle separation for maximum cut and binary quadratic optimization. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, Munich/Garching, Germany, September 9-11, 2019, volume 144 of LIPIcs, pages 63:1-63:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.63.
  28. Matteo Magnani, Barbora Micenková, and Luca Rossi. Combinatorial analysis of multiple networks. CoRR, abs/1303.4986, 2013. URL: https://arxiv.org/abs/1303.4986.
  29. Sven Mallach. On integer linear programs for treewidth based on perfect elimination orderings (extended version). Acta Informatica, 62(3):34, 2025. URL: https://doi.org/10.1007/S00236-025-00505-Y.
  30. Jan Mycielski. Sur le coloriage des graphs. Colloquium Mathematicae, 3(2):161-162, 1955. Google Scholar
  31. Assaf Natanzon, Ron Shamir, and Roded Sharan. A polynomial approximation algorithm for the minimum fill-in problem. SIAM J. Comput., 30(4):1067-1079, 2000. URL: https://doi.org/10.1137/S0097539798336073.
  32. George L. Nemhauser and Leslie E. Trotter Jr. Properties of vertex packing and independence system polyhedra. Math. Program., 6(1):48-61, 1974. URL: https://doi.org/10.1007/BF01580222.
  33. Stavros D. Nikolopoulos and Leonidas Palios. Detecting holes and antiholes in graphs. Algorithmica, 47(2):119-138, 2007. URL: https://doi.org/10.1007/S00453-006-1225-Y.
  34. Tiago P Peixoto. The netzschleuder network catalogue and repository. Zenodo, 2020. Google Scholar
  35. Dilson Lucas Pereira, Abilio Lucena, Alexandre Salles da Cunha, and Luidi Simonetti. Exact solution algorithms for the chordless cycle problem. INFORMS J. Comput., 34(4):1970-1986, 2022. URL: https://doi.org/10.1287/IJOC.2022.1164.
  36. Steffen Rebennack, Marcus Oswald, Dirk Oliver Theis, Hanna Seitz, Gerhard Reinelt, and Panos M. Pardalos. A branch and cut solver for the maximum stable set problem. J. Comb. Optim., 21(4):434-457, 2011. URL: https://doi.org/10.1007/S10878-009-9264-3.
  37. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  38. Paul Seymour. The origin of the notion of treewidth. https://cstheory.stackexchange.com/q/27317. Theoretical Computer Science Stack Exchange, 2014.
  39. Tom A. B. Snijders, Gerhard G. Van De Bunt, and Christian E. G. Steglich. Introduction to stochastic actor-based models for network dynamics. Social Networks, 32(1):44-60, 2010. URL: https://doi.org/10.1016/j.socnet.2009.02.004.
  40. Hisao Tamaki. Computing treewidth via exact and heuristic lists of minimal separators. In Ilias S. Kotsireas, Panos M. Pardalos, Konstantinos E. Parsopoulos, Dimitris Souravlias, and Arsenis Tsokas, editors, Analysis of Experimental Algorithms - Special Event, SEA² 2019, Kalamata, Greece, June 24-29, 2019, Revised Selected Papers, volume 11544 of Lecture Notes in Computer Science, pages 219-236. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-34029-2_15.
  41. Takeaki Uno and Hiroko Satoh. An efficient algorithm for enumerating chordless cycles and chordless paths. In Saso Dzeroski, Pance Panov, Dragi Kocev, and Ljupco Todorovski, editors, Discovery Science - 17th International Conference, DS 2014, Bled, Slovenia, October 8-10, 2014. Proceedings, volume 8777 of Lecture Notes in Computer Science, pages 313-324. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-11812-3_27.
  42. Malcolm P Young. The organization of neural systems in the primate cerebral cortex. Proceedings of the Royal Society of London. Series B: Biological Sciences, 252(1333):13-18, 1993. Google Scholar
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