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Roman Hitting Functions

Authors Henning Fernau , Kevin Mann

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Author Details

Henning Fernau
  • Fachbereich IV, Informatikwissenschaften, Universität Trier, Germany
Kevin Mann
  • Fachbereich IV, Informatikwissenschaften, Universität Trier, Germany

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Henning Fernau and Kevin Mann. Roman Hitting Functions. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.24

Abstract

Roman domination formalizes a military strategy going back to Constantine the Great. Here, armies are placed in different regions. A region is secured if there is at least one army in this region or there are two armies in one neighbored region. This simple strategy can be easily translated into a graph-theoretic question. The placement of armies is described by a function which maps each vertex to 0, 1 or 2. Such a function is called Roman dominating if each vertex with value 0 has a neighbor with value 2.
Roman domination is one of few examples where the related (so-called) extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting as it allows to establish polynomial-delay enumeration algorithms for listing minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets of a graph or (equivalently) if all hitting sets of a hypergraph can be enumerated with polynomial delay, or even in output-polynomial time. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain and study a new problem, called Roman Hitting Function, generalizing Roman Domination towards hypergraphs. This allows us to delineate the frontier of polynomial-delay enumerability. 
Our main focus is on the extension version of this problem, as this was the key problem when coping with Roman domination functions. While doing this, we find some conditions under which the Extension Roman Hitting Function problem is NP-complete. We then use parameterized complexity as a tool to get a better understanding of why Extension Roman Hitting Function behaves in this way. From an alternative perspective, we can say that we use the idea of parameterization to study the question what makes certain enumeration problems that difficult.
Also, we discuss another generalization of Extension Roman Domination, where both a lower and an upper bound on the sought minimal Roman domination function is provided. The additional upper bound makes the problem hard (again), and the applied parameterized complexity analysis (only) provides hardness results. 
Also from the viewpoint of Parameterized Complexity, the studies on extension problems are quite interesting as they provide more and more examples of parameterized problems complete for W[3], a complexity class with only very few natural members known five years ago.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • enumeration problems
  • polynomial delay
  • domination problems
  • hitting set
  • Roman domination

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References

  1. A. Aazami, J. Cheriyan, and K. R. Jampani. Approximation algorithms and hardness results for packing element-disjoint Steiner trees in planar graphs. Algorithmica, 63(1-2):425-456, 2012. URL: https://doi.org/10.1007/S00453-011-9540-3.
  2. F. N. Abu-Khzam, C. Bazgan, M. Chopin, and H. Fernau. Data reductions and combinatorial bounds for improved approximation algorithms. Journal of Computer and System Sciences, 82(3):503-520, 2016. URL: https://doi.org/10.1016/J.JCSS.2015.11.010.
  3. F. N. Abu-Khzam, H. Fernau, and K. Mann. Roman census: Enumerating and counting Roman dominating functions on graph classes. In J. Leroux, S. Lombardy, and D. Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science, MFCS, volume 272 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1-6:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.6.
  4. F. N. Abu-Khzam, H. Fernau, and K. Mann. Minimal Roman dominating functions: Extensions and enumeration. Algorithmica, 86:1862-1887, 2024. URL: https://doi.org/10.1007/s00453-024-01211-w.
  5. B. D. Acharya. Domination in hypergraphs. AKCE International Journal of Graphs and Combinatorics, 4(2):117-126, 2007. Google Scholar
  6. S. Bermudo and H. Fernau. Combinatorics for smaller kernels: The differential of a graph. Theoretical Computer Science, 562:330-345, 2015. URL: https://doi.org/10.1016/J.TCS.2014.10.007.
  7. S. Bermudo, H. Fernau, and J. M. Sigarreta. The differential and the Roman domination number of a graph. Applicable Analysis and Discrete Mathematics, 8:155-171, 2014. Google Scholar
  8. T. Bläsius, T. Friedrich, J. Lischeid, K. Meeks, and M. Schirneck. Efficiently enumerating hitting sets of hypergraphs arising in data profiling. In Algorithm Engineering and Experiments (ALENEX), pages 130-143. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975499.11.
  9. T. Bläsius, T. Friedrich, J. Lischeid, K. Meeks, and M. Schirneck. Efficiently enumerating hitting sets of hypergraphs arising in data profiling. Journal of Computer and System Sciences, 124:192-213, 2022. URL: https://doi.org/10.1016/J.JCSS.2021.10.002.
  10. T. Bläsius, T. Friedrich, and M. Schirneck. The complexity of dependency detection and discovery in relational databases. Theoretical Computer Science, 900:79-96, 2022. URL: https://doi.org/10.1016/J.TCS.2021.11.020.
  11. H. L. Bodlaender, C. Groenland, and M. Pilipczuk. On the complexity of problems on tree-structured graphs. Technical Report arXiv:2208.12543v3, ArXiv, Cornell University, 2022. URL: https://doi.org/10.48550/arXiv:2208.12543v3.
  12. F. Capelli and Y. Strozecki. Incremental delay enumeration: Space and time. Discrete Applied Mathematics, 268:179-190, 2019. URL: https://doi.org/10.1016/J.DAM.2018.06.038.
  13. K. Casel, H. Fernau, M. Khosravian Ghadikolaei, J. Monnot, and F. Sikora. On the complexity of solution extension of optimization problems. Theoretical Computer Science, 904:48-65, 2022. URL: https://doi.org/10.1016/j.tcs.2021.10.017.
  14. J. Chen and F. Zhang. On product covering in 3-tier supply chain models: Natural complete problems for W[3] and W[4]. Theoretical Computer Science, 363(3):278-288, 2006. URL: https://doi.org/10.1016/J.TCS.2006.07.016.
  15. Y. Chen, J. Flum, and M. Grohe. An analysis of the W^*-hierarchy. The Journal of Symbolic Logic, 72(2):513-534, 2007. URL: https://doi.org/10.2178/JSL/1185803622.
  16. E. J. Cockayne, P. A. Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi. Roman domination in graphs. Discrete Mathematics, 278:11-22, 2004. URL: https://doi.org/10.1016/J.DISC.2003.06.004.
  17. N. Creignou, M. Kröll, R. Pichler, S. Skritek, and H. Vollmer. A complexity theory for hard enumeration problems. Discrete Applied Mathematics, 268:191-209, 2019. URL: https://doi.org/10.1016/J.DAM.2019.02.025.
  18. R. G. Downey and M. R. Fellows. Threshold dominating sets and an improved characterization of W[2]. Theoretical Computer Science, 209(1-2):123-140, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00101-1.
  19. R. G. Downey and M. R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  20. P. A. Dreyer. Applications and Variations of Domination in Graphs. PhD thesis, Rutgers University, New Jersey, USA, 2000. Google Scholar
  21. T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278-1304, 1995. URL: https://doi.org/10.1137/S0097539793250299.
  22. H. Fernau. Roman Domination: a parameterized perspective. International Journal of Computer Mathematics, 85:25-38, 2008. URL: https://doi.org/10.1080/00207160701374376.
  23. J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  24. A. Gainer-Dewar and P. Vera-Licona. The minimal hitting set generation problem: Algorithms and computation. SIAM Journal of Discrete Mathematics, 31(1):63-100, 2017. URL: https://doi.org/10.1137/15M1055024.
  25. G. Gogic, C. H. Papadimitriou, and M. Sideri. Incremental recompilation of knowledge. Journal of Artificial Intelligence Research, 8:23-37, 1998. URL: https://doi.org/10.1613/JAIR.380.
  26. J. Guo, F. Hüffner, and R. Niedermeier. A structural view on parameterizing problems: distance from triviality. In R. Downey, M. Fellows, and F. Dehne, editors, International Workshop on Parameterized and Exact Computation IWPEC 2004, volume 3162 of LNCS, pages 162-173. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_15.
  27. T. W. Haynes, S.T. Hedetniemi, and M. A. Henning, editors. Topics in Domination in Graphs, volume 64 of Developments in Mathematics. Springer, 2020. Google Scholar
  28. M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. Enumeration of minimal dominating sets and variants. In O. Owe, M. Steffen, and J. A. Telle, editors, Fundamentals of Computation Theory - 18th International Symposium, FCT, volume 6914 of LNCS, pages 298-309. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22953-4_26.
  29. M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal of Discrete Mathematics, 28(4):1916-1929, 2014. URL: https://doi.org/10.1137/120862612.
  30. M. Liedloff, T. Kloks, J. Liu, and S.-L. Peng. Efficient algorithms for Roman domination on some classes of graphs. Discrete Applied Mathematics, 156(18):3400-3415, 2008. URL: https://doi.org/10.1016/J.DAM.2008.01.011.
  31. K. Mann and H. Fernau. Perfect Roman domination: Aspects of enumeration and parameterization. In A. A. Rescigno and U. Vaccaro, editors, Combinatorial Algorithms (Proceeding 35th International Workshop on Combinatorial Algorithms IWOCA), volume 14764 of LNCS, pages 354-368. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-63021-7_27.
  32. A. Mary. Énumération des dominants minimaux d'un graphe. PhD thesis, LIMOS, Université Blaise Pascal, Clermont-Ferrand, France, November 2013. Google Scholar
  33. A. Mary and Y. Strozecki. Efficient enumeration of solutions produced by closure operations. Discrete Mathematics & Theoretical Computer Science, 21(3), 2019. URL: https://doi.org/10.23638/DMTCS-21-3-22.
  34. C. Padamutham and V. S. R. Palagiri. Algorithmic aspects of Roman domination in graphs. Journal of Applied Mathematics and Computing, 64:89-102, 2020. URL: https://doi.org/10.1007/S12190-020-01345-4.
  35. A. Pagourtzis, P. Penna, K. Schlude, K. Steinhöfel, D. S. Taylor, and P. Widmayer. Server placements, Roman domination and other dominating set variants. In R. A. Baeza-Yates, U. Montanari, and N. Santoro, editors, Foundations of Information Technology in the Era of Networking and Mobile Computing, IFIP 17^th World Computer Congress - TC1 Stream / 2^nd IFIP International Conference on Theoretical Computer Science IFIP TCS, pages 280-291. Kluwer, 2002. Also available as Technical Report 365, ETH Zürich, Institute of Theoretical Computer Science, 10/2001. URL: https://doi.org/10.1007/978-0-387-35608-2_24.
  36. R. C. Read and R. E. Tarjan. Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks, 5:237-252, 1975. URL: https://doi.org/10.1002/NET.1975.5.3.237.
  37. Y. Strozecki. Enumeration complexity. EATCS Bulletin, 129, 2019. Google Scholar
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