Christian Komusiewicz
Abstract
For a graph class Π, the Π-Vertex Deletion problem has as input an undirected graph G = (V,E) and an integer k and asks whether there is a set of at most k vertices that can be deleted from G such that the resulting graph is a member of Π. By a classic result of Lewis and Yannakakis [17], Π-Vertex Deletion is NP-hard for all hereditary properties Π. We adapt the original NP-hardness construction to show that under the exponential time hypothesis (ETH), tight complexity results can be obtained. We show that Π-Vertex Deletion does not admit a 2o(n)-time algorithm where n is the number of vertices in G. We also obtain a dichotomy for running time bounds that include the number m of edges in the input graph. On the one hand, if Π contains all edgeless graphs, then there is no 2o(n+m)-time algorithm for Π-Vertex Deletion. On the other hand, if there is a fixed edgeless graph that is not contained in Π and containment in Π can be determined in 2O(n) time or 2o(m) time, then Π-Vertex Deletion can be solved in 2O(√m)+O(n) or 2o(m)+O(n) time, respectively. We also consider restrictions on the domain of the input graph G. For example, we obtain that Π-Vertex Deletion cannot be solved in 2o(√n) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.
- N. R. Aravind, R. B. Sandeep, and N. Sivadasan. 2017. Dichotomy results on the hardness of H-free edge modification problems. SIAM Journal on Discrete Mathematics 31, 1, 542--561.Google Scholar
Cross Ref
- Balabhaskar Balasundaram, Sergiy Butenko, and Illya V. Hicks. 2011. Clique relaxations in social network analysis: The maximum k-plex problem. Operations Research 59, 1, 133--142. Google ScholarDigital Library
- Sebastian Böcker and Jan Baumbach. 2013. Cluster editing. In The Nature of Computation, Logic, Algorithms, Applications. Lecture Notes in Computer Science, Vol. 7921. Springer, 33--44.Google Scholar
- Kevin Buchin, Maike Buchin, Jaroslaw Byrka, Martin Nöllenburg, Yoshio Okamoto, Rodrigo I. Silveira, and Alexander Wolff. 2012. Drawing (complete) binary tanglegrams—hardness, approximation, fixed-parameter tractability. Algorithmica 62, 1--2, 309--332. Google ScholarDigital Library
- Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer, Heidelberg, Germany. Google ScholarDigital Library
- Pål Grønås Drange, Fedor V. Fomin, Michal Pilipczuk, and Yngve Villanger. 2015. Exploring the subexponential complexity of completion problems. ACM Transactions on Computation Theory 7, 4, 14. Google ScholarDigital Library
- Michael R. Fellows, Jiong Guo, Hannes Moser, and Rolf Niedermeier. 2011. A complexity dichotomy for finding disjoint solutions of vertex deletion problems. ACM Transactions on Computation Theory 2, 2, 5. Google ScholarDigital Library
- Bernold Fiedler, Atsushi Mochizuki, Gen Kurosawa, and Daisuke Saito. 2013. Dynamics and control at feedback vertex sets. I: Informative and determining nodes in regulatory networks. Journal of Dynamics and Differential Equations 25, 3, 563--604.Google ScholarCross Ref
- Fedor V. Fomin and Dieter Kratsch. 2010. Exact Exponential Algorithms. Springer, Heidelberg, Germany. Google ScholarDigital Library
- Michael R. Garey, David S. Johnson, and Larry J. Stockmeyer. 1976. Some simplified NP-complete graph problems. Theoretical Computer Science 1, 3, 237--267.Google ScholarCross Ref
- Alexander Grigoriev and René Sitters. 2010. Connected feedback vertex set in planar graphs. In Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science, Vol. 5911. Springer, 143--153. Google ScholarDigital Library
- Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. 2007. Parameterized complexity of vertex cover variants. Theory of Computing Systems 41, 3, 501--520. Google ScholarDigital Library
- Jiong Guo and Yash Raj Shrestha. 2014. Complexity of disjoint π-vertex deletion for disconnected forbidden subgraphs. Journal of Graph Algorithms and Applications 18, 4, 603--631.Google ScholarCross Ref
- Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. 2001. Which problems have strongly exponential complexity?Journal of Computer and System Sciences 63, 4, 512--530. Google ScholarDigital Library
- David S. Johnson and Mario Szegedy. 1999. What are the least tractable instances of max independent set? In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’99). 927--928. Google ScholarDigital Library
- Christian Komusiewicz. 2016. Multivariate algorithmics for finding cohesive subnetworks. Algorithms 9, 1, 21.Google ScholarCross Ref
- John M. Lewis and Mihalis Yannakakis. 1980. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences 20, 2, 219--230.Google ScholarCross Ref
- Kun Liu and Evimaria Terzi. 2008. Towards identity anonymization on graphs. In Proceedings of the ACM International Conference on Management of Data (SIGMOD’08). ACM, New York, NY, 93--106. Google ScholarDigital Library
- Xianliang Liu, Hongliang Lu, Wei Wang, and Weili Wu. 2013. PTAS for the minimum k-path connected vertex cover problem in unit disk graphs. Journal of Global Optimization 56, 2, 449--458. Google ScholarDigital Library
- Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. 2011. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS 105, 41--72.Google Scholar
- Dániel Marx. 2013. The square root phenomenon in planar graphs. In Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, Vol. 7966. Springer, 28.Google Scholar
- Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, Saket Saurabh, and Somnath Sikdar. 2012. FPT algorithms for connected feedback vertex set. Journal of Combinatorial Optimization 24, 2, 131--146. Google ScholarDigital Library
- André Nichterlein. 2014. Degree-Constrained Editing of Small-Degree Graphs. Ph.D. Dissertation. TU Berlin.Google Scholar
- Mihalis Yannakakis. 1981. Edge-deletion problems. SIAM Journal on Computing 10, 2 297--309.Google ScholarCross Ref
Index Terms
- Tight Running Time Lower Bounds for Vertex Deletion Problems
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