Problems in NP Can Admit Double-Exponential Lower Bounds When Parameterized by Treewidth or Vertex Cover

Authors Florent Foucaud , Esther Galby , Liana Khazaliya , Shaohua Li , Fionn Mc Inerney , Roohani Sharma , Prafullkumar Tale



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.66.pdf
  • Filesize: 1.15 MB
  • 19 pages

Document Identifiers

Author Details

Florent Foucaud
  • Université Clermont Auvergne, CNRS, Mines Saint-Étienne, Clermont Auvergne INP, LIMOS, 63000 Clermont-Ferrand, France
Esther Galby
  • Department of Computer Science and Engineering, Chalmers University of Technology, Gothenburg, Sweden
  • University of Gothenburg, Sweden
Liana Khazaliya
  • Technische Universität Wien, Austria
Shaohua Li
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Fionn Mc Inerney
  • Technische Universität Wien, Austria
Roohani Sharma
  • University of Bergen, Norway
Prafullkumar Tale
  • Indian Institute of Science Education and Research Pune, India

Cite AsGet BibTex

Florent Foucaud, Esther Galby, Liana Khazaliya, Shaohua Li, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Problems in NP Can Admit Double-Exponential Lower Bounds When Parameterized by Treewidth or Vertex Cover. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 66:1-66:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.66

Abstract

Treewidth serves as an important parameter that, when bounded, yields tractability for a wide class of problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and Quantified SAT or, more generally, Quantified CSP, are fixed-parameter tractable parameterized by the treewidth {of the input’s (primal) graph} plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], respectively. The algorithms generated by these (meta-)results have running times whose dependence on treewidth is a tower of exponents. A conditional lower bound by Fichte, Hecher, and Pfandler [LICS 2020] shows that, for Quantified SAT, the height of this tower is equal to the number of quantifier alternations. These types of lower bounds, which show that at least double-exponential factors in the running time are necessary, exhibit the extraordinary level of computational hardness for such problems, and are rare in the current literature: there are only a handful of such lower bounds (for treewidth and vertex cover parameterizations) and all of them are for problems that are #NP-complete, Σ₂^p-complete, Π₂^p-complete, or complete for even higher levels of the polynomial hierarchy. Our results demonstrate, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to achieve double-exponential lower bounds: we derive double-exponential lower bounds in the treewidth (tw) and the vertex cover number (vc), for natural, important, and well-studied NP-complete graph problems. Specifically, we design a technique to obtain such lower bounds and show its versatility by applying it to three different problems: Metric Dimension, Strong Metric Dimension, and Geodetic Set. We prove that these problems do not admit 2^{2^o(tw)}⋅n^𝒪(1)-time algorithms, even on bounded diameter graphs, unless the ETH fails (here, n is the number of vertices in the graph). In fact, for Strong Metric Dimension, the double-exponential lower bound holds even for the vertex cover number. We further complement all our lower bounds with matching (and sometimes non-trivial) upper bounds. For the conditional lower bounds, we design and use a novel, yet simple technique based on Sperner families of sets. We believe that the amenability of our technique will lead to obtaining such lower bounds for many other problems in NP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • ETH-based Lower Bounds
  • Double-Exponential Lower Bounds
  • Kernelization
  • Vertex Cover
  • Treewidth
  • Diameter
  • Metric Dimension
  • Strong Metric Dimension
  • Geodetic Sets

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. A. Achilleos, M. Lampis, and V. Mitsou. Parameterized modal satisfiability. Algorithmica, 64(1):38-55, 2012. Google Scholar
  2. J. Ahn, L. Jaffke, O. Kwon, and P. T. Lima. Well-partitioned chordal graphs. Discrete Mathematics, 345(10):112985, 2022. Google Scholar
  3. F. Barbero, L. Isenmann, and J. Thiebaut. On the distance identifying set meta-problem and applications to the complexity of identifying problems on graphs. Algorithmica, 82(8):2243-2266, 2020. URL: https://doi.org/10.1007/S00453-020-00674-X.
  4. R. Belmonte, F. V. Fomin, P. A. Golovach, and M. S. Ramanujan. Metric dimension of bounded tree-length graphs. SIAM J. Discrete Math., 31(2):1217-1243, 2017. Google Scholar
  5. B. Bergougnoux, O. Defrain, and F. Mc Inerney. Enumerating minimal solution sets for metric graph problems. In Proc. of the 50th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2024), Lecture Notes in Computer Science. Springer, 2024. Google Scholar
  6. E. Bonnet and N. Purohit. Metric dimension parameterized by treewidth. Algorithmica, 83:2606-2633, 2021. Google Scholar
  7. N. Bousquet, Q. Deschamps, and A. Parreau. Metric dimension parameterized by treewidth in chordal graphs. In Proc. of the 49th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2023), volume 14093 of Lecture Notes in Computer Science, pages 130-142. Springer, 2023. Google Scholar
  8. D. Chakraborty, S. Das, F. Foucaud, H. Gahlawat, D. Lajou, and B. Roy. Algorithms and complexity for geodetic sets on planar and chordal graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020), volume 181 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  9. D. Chakraborty, F. Foucaud, H. Gahlawat, S. K. Ghosh, and B. Roy. Hardness and approximation for the geodetic set problem in some graph classes. In Proc. of the 6th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2020), volume 12016 of Lecture Notes in Computer Science, pages 102-115, Cham, 2020. Springer. Google Scholar
  10. D. Chakraborty, F. Foucaud, D. Majumdar, and P. Tale. Tight (double) exponential bounds for identification problems: Locating-dominating set and test cover, 2024. URL: https://arxiv.org/abs/2402.08346.
  11. J. Chalopin, V. Chepoi, F. Mc Inerney, and S. Ratel. Non-clashing teaching maps for balls in graphs. Arxiv:2309.02876, 2023. Google Scholar
  12. H. Chen. Quantified constraint satisfaction and bounded treewidth. In Proc. of the 16th Eureopean Conference on Artificial Intelligence, ECAI'2004, pages 161-165. IOS Press, 2004. Google Scholar
  13. B. Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. Google Scholar
  14. M. Cygan, D. Marx, M. Pilipczuk, and M. Pilipczuk. Hitting forbidden subgraphs in graphs of bounded treewidth. Inf. Comput., 256(C):62-82, 2017. Google Scholar
  15. M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. M. M. van Rooij, and J. O. Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011), pages 150-159, 2011. Google Scholar
  16. M. Cygan, M. Pilipczuk, and M. Pilipczuk. Known algorithms for edge clique cover are probably optimal. SIAM J. Comput., 45(1):67-83, 2016. URL: https://doi.org/10.1137/130947076.
  17. B. DasGupta and N. Mobasheri. On optimal approximability results for computing the strong metric dimension. Discrete Applied Math., 221:18-24, 2017. Google Scholar
  18. J. Díaz, O. Pottonen, M. J. Serna, and E. J. van Leeuwen. Complexity of metric dimension on planar graphs. J. Comput. Syst. Sci., 83(1):132-158, 2017. Google Scholar
  19. M. C. Dourado, F. Protti, D. Rautenbach, and J. L. Szwarcfiter. Some remarks on the geodetic number of a graph. Discrete Mathematics, 310(4):832-837, 2010. Google Scholar
  20. T. Ekim, A. Erey, P. Heggernes, P. van’t Hof, and D. Meister. Computing minimum geodetic sets of proper interval graphs. In Proc. of the 10th Latin American Symposium on Theoretical Informatics (LATIN 2012), volume 7256 of Lecture Notes in Computer Science, pages 279-290. Springer, 2012. Google Scholar
  21. D. Eppstein. Metric dimension parameterized by max leaf number. Journal of Graph Algorithms and Applications, 19(1):313-323, 2015. Google Scholar
  22. L. Epstein, A. Levin, and G. J. Woeginger. The (weighted) metric dimension of graphs: Hard and easy cases. Algorithmica, 72(4):1130-1171, 2015. Google Scholar
  23. J. K. Fichte, M. Hecher, M. Morak, P. Thier, and S. Woltran. Solving projected model counting by utilizing treewidth and its limits. Artif. Intell., 314:103810, 2023. Google Scholar
  24. J. K. Fichte, M. Hecher, M. Morak, and S. Woltran. Exploiting treewidth for projected model counting and its limits. In Theory and Applications of Satisfiability Testing - SAT 2018 - 21st International Conference, SAT 2018, Proc., volume 10929 of Lecture Notes in Computer Science, pages 165-184. Springer, 2018. Google Scholar
  25. J. K. Fichte, M. Hecher, and A. Pfandler. Lower bounds for QBFs of bounded treewidth. In LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 410-424. ACM, 2020. Google Scholar
  26. J. Focke, F. Frei, S. Li, D. Marx, P. Schepper, R. Sharma, and K. Wegrzycki. Hitting meets packing: How hard can it be? CoRR, abs/2402.14927, 2024. URL: https://doi.org/10.48550/arXiv.2402.14927.
  27. F. V. Fomin, P. A. Golovach, D. Lokshtanov, S. Saurabh, and M. Zehavi. Clique-width III: hamiltonian cycle and the odd case of graph coloring. ACM Trans. Algorithms, 15(1):9:1-9:27, 2019. Google Scholar
  28. F. Foucaud, G. B. Mertzios, R. Naserasr, A. Parreau, and P. Valicov. Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity. Algorithmica, 78(3):914-944, 2017. Google Scholar
  29. E. Galby, L. Khazaliya, F. Mc Inerney, R. Sharma, and P. Tale. Metric dimension parameterized by feedback vertex set and other structural parameters. SIAM J. Discrete Math., 37(4):2241-2264, 2023. Google Scholar
  30. M. R. Garey and D. S. Johnson. Computers and Intractability - A guide to NP-completeness. W.H. Freeman and Company, 1979. Google Scholar
  31. T. Gima, T. Hanaka, M. Kiyomi, Y. Kobayashi, and Y. Otachi. Exploring the gap between treedepth and vertex cover through vertex integrity. Theoretical Computer Science, 918:60-76, 2022. URL: https://doi.org/10.1016/J.TCS.2022.03.021.
  32. T. Hanaka, H. Köhler, and M. Lampis. Core stability in additively separable hedonic games of low treewidth, 2024. URL: https://arxiv.org/abs/2402.10815.
  33. F. Harary, E. Loukakis, and C. Tsouros. The geodetic number of a graph. Mathematical and Computer Modelling, 17(11):89-95, 1993. Google Scholar
  34. F. Harary and R. A. Melter. On the metric dimension of a graph. Ars Combinatoria, 2:191-195, 1976. Google Scholar
  35. S. Hartung and A. Nichterlein. On the parameterized and approximation hardness of metric dimension. In Proc. of the 28th Conference on Computational Complexity, CCC 2013, pages 266-276. IEEE Computer Society, 2013. Google Scholar
  36. A. Ian. Combinatorics of Finite Sets. Oxford University Press, 1987. Google Scholar
  37. K. Jansen, KM. Klein, and A. Lassota. The double exponential runtime is tight for 2-stage stochastic ILPs. Math. Program., 197:1145-1172, 2023. Google Scholar
  38. M. M. Kanté and L. Nourine. Polynomial time algorithms for computing a minimum hull set in distance-hereditary and chordal graphs. SIAM J. Discrete Math., 30(1):311-326, 2016. Google Scholar
  39. L. Kellerhals and T. Koana. Parameterized complexity of geodetic set. Journal of Graph Algorithms and Applications, 26(4):401-419, 2022. Google Scholar
  40. T. Kloks. Treewidth, Computations and Approximations. Springer, 1994. Google Scholar
  41. D. Knop, M. Pilipczuk, and M. Wrochna. Tight complexity lower bounds for integer linear programming with few constraints. ACM Trans. Comput. Theory, 12(3):19:1-19:19, 2020. Google Scholar
  42. T. Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS 2021), pages 184-192, 2022. Google Scholar
  43. L. Kowalik, A. Lassota, K. Majewski, M. Pilipczuk, and M. Sokołowski. Detecting points in integer cones of polytopes is double-exponentially hard. In 2024 Symposium on Simplicity in Algorithms (SOSA), pages 279-285, 2024. Google Scholar
  44. M. Künnemann, F. Mazowiecki, L. Schütze, H. Sinclair-Banks, and K. Węgrzycki. Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 131:1-131:20, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  45. D. Kuziak, M. L. Puertas, J. A. Rodríguez-Velázquez, and I. G. Yero. Strong resolving graphs: The realization and the characterization problems. Discrete Applied Math., 236:270-287, 2018. Google Scholar
  46. M. Lampis, N. Melissinos, and M. Vasilakis. Parameterized max min feedback vertex set. In 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023, volume 272 of LIPIcs, pages 62:1-62:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  47. M. Lampis, S. Mengel, and V. Mitsou. QBF as an alternative to courcelle’s theorem. In Theory and Applications of Satisfiability Testing - SAT 2018 - 21st International Conference, SAT 2018, volume 10929 of Lecture Notes in Computer Science, pages 235-252. Springer, 2018. Google Scholar
  48. M. Lampis and V. Mitsou. Treewidth with a quantifier alternation revisited. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), volume 89 of LIPIcs, pages 26:1-26:12, 2017. Google Scholar
  49. S. Li and M. Pilipczuk. Hardness of metric dimension in graphs of constant treewidth. Algorithmica, 84(11):3110-3155, 2022. Google Scholar
  50. D. Lokshtanov, D. Marx, and S. Saurabh. Slightly superexponential parameterized problems. SIAM J. Comput., 47(3):675-702, 2018. Google Scholar
  51. D. Lokshtanov, S. Saurabh, S. Suri, and J. Xue. An ETH-tight algorithm for multi-team formation. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2021, volume 213 of LIPIcs, pages 28:1-28:9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  52. D. Marx and V. Mitsou. Double-exponential and triple-exponential bounds for choosability problems parameterized by treewidth. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55 of LIPIcs, pages 28:1-28:15, 2016. Google Scholar
  53. M. Mezzini. Polynomial time algorithm for computing a minimum geodetic set in outerplanar graphs. Theoretical Computer Science, 745:63-74, 2018. Google Scholar
  54. O. R. Oellermann and J. Peters-Fransen. The strong metric dimension of graphs and digraphs. Discrete Applied Mathematics, 155(3):356-364, 2007. Google Scholar
  55. M. Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proc., volume 6907 of Lecture Notes in Computer Science, pages 520-531. Springer, 2011. Google Scholar
  56. M. Pilipczuk and M. Sorge. A double exponential lower bound for the distinct vectors problem. Discret. Math. Theor. Comput. Sci., 22(4), 2020. Google Scholar
  57. A. Sebő and E. Tannier. On metric generators of graphs. Mathematics of Operations Research, 29(2):383-393, 2004. Google Scholar
  58. P. J. Slater. Leaves of trees. In Proc. of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 549-559. Congressus Numerantium, No. XIV. Utilitas Mathematica, 1975. Google Scholar
  59. P. Tale. Double exponential lower bound for telephone broadcast, 2024. URL: https://arxiv.org/abs/2403.03501.