research-article

Exploiting $c$-Closure in Kernelization Algorithms for Graph Problems

Authors:

Tomohiro Koana,

Christian Komusiewicz,

Frank SommerAuthors Info & Claims

SIAM Journal on Discrete Mathematics, Volume 36, Issue 4

Pages 2798 - 2821

https://doi.org/10.1137/21M1449476

Published: 01 January 2022 Publication History

Abstract

A graph is $c$-closed if every pair of vertices with at least $c$ common neighbors is adjacent. The $c$-closure of a graph $G$ is the smallest number $c$ such that $G$ is $c$-closed. Fox et al. [SIAM J. Comput., 49 (2020), pp. 448--464] defined $c$-closure and investigated it in the context of clique enumeration. We show that $c$-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size $k^{\mathcal{O}(c)}$, that Induced Matching admits a kernel with $\mathcal{O}(c^7 k^{8})$ vertices, and that Irredundant Set admits a kernel with $\mathcal{O}(c^{5/2} k^3)$ vertices. As we show, our kernelizations exploit the fact that $c$-closed graphs have polynomially bounded Ramsey numbers.

References

[1]

F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, and W. H. Suters, Crown structures for vertex cover kernelization, Theory Comput. Syst., 41 (2007), pp. 411--430.

[2]

V. E. Alekseev, D. V. Korobitsyn, and V. V. Lozin, Boundary classes of graphs for the dominating set problem, Discrete Math., 285 (2004), pp. 1--6.

[3]

N. Alon and S. Gutner, Linear time algorithms for finding a dominating set of fixed size in degenerated graphs, Algorithmica, 54 (2009), pp. 544--556.

[4]

R. Bar-Yehuda and S. Even, A local-ratio theorem for approximating the weighted vertex cover problem, in Proceedings of the 9th International Workshop on Graph-Theoretic Concepts in Computer Science (WG '83), Universitätsverlag Rudolf Trauner, Linz, 1983, pp. 17--28.

[5]

B. Behera, E. Husić, S. Jain, T. Roughgarden, and C. Seshadhri, FPT algorithms for finding near-cliques in c-closed graphs, in Proceedings of the 13th Innovations in Theoretical Computer Science Conference (ITCS '22), LIPIcs. Leibniz Int. Proc. Inform. 215, Schloss Dagstuhl. Leibniz-Zent. Inform., 2022, 17.

[6]

L. Cai, Parameterized complexity of cardinality constrained optimization problems, Comput. J., 51 (2008), pp. 102--121.

[7]

J. Chen, I. A. Kanj, and W. Jia, Vertex cover: Further observations and further improvements, J. Algorithms, 41 (2001), pp. 280--301.

[8]

M. Chlebík and J. Chlebíková, Crown reductions for the minimum weighted vertex cover problem, Discrete Appl. Math., 156 (2008), pp. 292--312.

[9]

B. Chor, M. Fellows, and D. W. Juedes, Linear kernels in linear time, or how to save $k$ colors in $O(n^2)$ steps, in Proceedings of the 30th International Workshop on Graph-Theoretic Concepts in Computer Science (WG '04), Lecture Notes in Comput. Sci. 3353, Springer, 2004, pp. 257--269.

[10]

M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh, Parameterized Algorithms, Springer, 2015.

[11]

M. Cygan, F. Grandoni, and D. Hermelin, Tight kernel bounds for problems on graphs with small degeneracy, ACM Trans. Algorithms, 13 (2017), 43.

[12]

K. Dabrowski, M. Demange, and V. V. Lozin, New results on maximum induced matchings in bipartite graphs and beyond, Theoret. Comput. Sci., 478 (2013), pp. 33--40.

[13]

H. Dell and D. Marx, Kernelization of packing problems, in Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2012, pp. 68--81, https://doi.org/10.1137/1.9781611973099.6.

[14]

H. Dell and D. van Melkebeek, Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses, J. ACM, 61 (2014), 23.

[15]

R. G. Downey and M. R. Fellows, Fundamentals of Parameterized Complexity, Texts Comput. Sci., Springer, 2013.

[16]

R. G. Downey, M. R. Fellows, and V. Raman, The complexity of irredundant sets parameterized by size, Discrete Appl. Math., 100 (2000), pp. 155--167.

[17]

P. Erdös, Some remarks on the theory of graphs, Bull. Amer. Math. Soc., 53 (1947), pp. 292--294.

[18]

R. Erman, L. Kowalik, M. Krnc, and T. Waleń, Improved induced matchings in sparse graphs, Discrete Appl. Math., 158 (2010), pp. 1994--2003.

[19]

J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein, Finding cliques in social networks: A new distribution-free model, SIAM J. Comput., 49 (2020), pp. 448--464, https://doi.org/10.1137/18M1210459.

[20]

S. Garg and G. Philip, Raising the bar for vertex cover: Fixed-parameter tractability above a higher guarantee, in Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2016, pp. 1152--1166, https://doi.org/10.1137/1.9781611974331.ch80.

[21]

P. A. Golovach and Y. Villanger, Parameterized complexity for domination problems on degenerate graphs, in Proceedings of the 34th International Workshop on Graph-Theoretic Concepts in Computer Science (WG '08), Lecture Notes in Comput. Sci. 5344, Springer, 2008, pp. 195--205.

[22]

D. Hermelin and X. Wu, Weak compositions and their applications to polynomial lower bounds for kernelization, in Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2012, pp. 104--113, https://doi.org/10.1137/1.9781611973099.9.

[23]

R. Impagliazzo, R. Paturi, and F. Zane, Which problems have strongly exponential complexity?, J. Comput. System Sci., 63 (2001), pp. 512--530.

[24]

S. Jukna, Extremal Combinatorics: With Applications in Computer Science, 2nd ed., Texts Theoret. Comput. Sci. EATCS Ser., Springer, 2011.

[25]

L. Kanesh, J. Madathil, S. Roy, A. Sahu, and S. Saurabh, Further exploiting c-closure for FPT algorithms and kernels for domination problems, in Proceedings of the 39th International Symposium on Theoretical Aspects of Computer Science (STACS '22), LIPIcs. Leibniz Int. Proc. Inform. 219, Schloss Dagstuhl. Leibniz-Zent. Inform., 2022, 39.

[26]

I. A. Kanj, M. J. Pelsmajer, M. Schaefer, and G. Xia, On the induced matching problem, J. Comput. Syst. Sci., 77 (2011), pp. 1058--1070.

[27]

T. Koana, C. Komusiewicz, A. Nichterlein, and F. Sommer, Covering many (or few) edges with $k$ vertices in sparse graphs, in Proceedings of the 39th International Symposium on Theoretical Aspects of Computer Science (STACS '22), LIPIcs. Leibniz Int. Proc. Inform. 219, Schloss Dagstuhl. Leibniz-Zent. Inform., 2022, 42.

[28]

T. Koana, C. Komusiewicz, and F. Sommer, Computing dense and sparse subgraphs of weakly closed graphs, in Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC '20), LIPIcs. Leibniz Int. Proc. Inform. 181, Schloss Dagstuhl. Leibniz-Zent. Inform., 2020, 20.

[29]

T. Koana, C. Komusiewicz, and F. Sommer, Exploiting $c$-closure in kernelization algorithms for graph problems, in Proceedings of the 28th Annual European Symposium on Algorithms (ESA '20), LIPIcs. Leibniz Int. Proc. Inform. 173, Schloss Dagstuhl Leibniz-Zent. Inform., 2020, 65.

[30]

T. Koana, C. Komusiewicz, and F. Sommer, Essentially tight kernels for (weakly) closed graphs, in Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC '21), LIPIcs. Leibniz Int. Proc. Inform. 212, Schloss Dagstuhl Leibniz-Zent. Inform., 2021, 35.

[31]

T. Koana and A. Nichterlein, Detecting and enumerating small induced subgraphs in c-closed graphs, Discrete Appl. Math., 302 (2021), pp. 198--207.

[32]

C. Komusiewicz and M. Sorge, An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems, Discrete Appl. Math., 193 (2015), pp. 145--161.

[33]

D. Lokshtanov, D. Marx, and S. Saurabh, Slightly superexponential parameterized problems, SIAM J. Comput., 47 (2018), pp. 675--702, https://doi.org/10.1137/16M1104834.

[34]

D. Lokshtanov and V. Surianarayanan, Dominating set in weakly closed graphs is fixed parameter tractable, in Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS '21), LIPIcs. Leibniz Int. Proc. Inform. 213, Schloss Dagstuhl Leibniz-Zent. Inform., 2021, 29.

[35]

H. Moser and S. Sikdar, The parameterized complexity of the induced matching problem, Discrete Appl. Math., 157 (2009), pp. 715--727.

[36]

H. Moser and D. M. Thilikos, Parameterized complexity of finding regular induced subgraphs, J. Discrete Algorithms, 7 (2009), pp. 181--190.

[37]

G. L. Nemhauser and L. E. Trotter, Vertex packings: Structural properties and algorithms, Math. Programming, 8 (1975), pp. 232--248.

[38]

G. Philip, V. Raman, and S. Sikdar, Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond, ACM Trans. Algorithms, 9 (2012), 11.

[39]

V. Raman and S. Saurabh, Short cycles make W-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles, Algorithmica, 52 (2008), pp. 203--225.

[40]

J. A. Telle and Y. Villanger, FPT algorithms for domination in biclique-free graphs, in Proceedings of the 20th Annual European Symposium on Algorithms (ESA '12), Lecture Notes in Comput. Sci. 7501, Springer, 2012, pp. 802--812.

Cited By

Grandi UKanesh LLisowski GSridharan RTurrini PWilliams BChen YNeville J(2023)Identifying and eliminating majority illusion in social networksProceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v37i4.25634(5062-5069)Online publication date: 7-Feb-2023
https://dl.acm.org/doi/10.1609/aaai.v37i4.25634
Koana TKomusiewicz CSommer F(2023)Computing Dense and Sparse Subgraphs of Weakly Closed GraphsAlgorithmica10.1007/s00453-022-01090-z85:7(2156-2187)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1007/s00453-022-01090-z
Koana TKomusiewicz CSommer F(2023)Essentially Tight Kernels for (Weakly) Closed GraphsAlgorithmica10.1007/s00453-022-01088-785:6(1706-1735)Online publication date: 1-Jun-2023
https://dl.acm.org/doi/10.1007/s00453-022-01088-7

Index Terms

Exploiting $c$-Closure in Kernelization Algorithms for Graph Problems
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Graph enumeration
  2. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

Recommendations

Essentially Tight Kernels for (Weakly) Closed Graphs
Abstract
We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number $γ$ (Fox et al. SIAM J Comput 49(... $^{}$ $^{}$ $^{}$ $^{}$ $^{}$ $^{}$ $^{}$
Kernelization of Graph Hamiltonicity: Proper $H$-Graphs

We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choice of parameterization is strongly influenced by the work of ...
Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs
Abstract
We study algorithmic properties of the graph class $C H O R D A L - k e$ , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental ... $^{\sqrt{}}^{}$

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Discrete Mathematics

SIAM Journal on Discrete Mathematics Volume 36, Issue 4

DOI:10.1137/sjdmec.36.4

Issue’s Table of Contents

© 2022, Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2022

Author Tags

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 04 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Grandi UKanesh LLisowski GSridharan RTurrini PWilliams BChen YNeville J(2023)Identifying and eliminating majority illusion in social networksProceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v37i4.25634(5062-5069)Online publication date: 7-Feb-2023
https://dl.acm.org/doi/10.1609/aaai.v37i4.25634
Koana TKomusiewicz CSommer F(2023)Computing Dense and Sparse Subgraphs of Weakly Closed GraphsAlgorithmica10.1007/s00453-022-01090-z85:7(2156-2187)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1007/s00453-022-01090-z
Koana TKomusiewicz CSommer F(2023)Essentially Tight Kernels for (Weakly) Closed GraphsAlgorithmica10.1007/s00453-022-01088-785:6(1706-1735)Online publication date: 1-Jun-2023
https://dl.acm.org/doi/10.1007/s00453-022-01088-7

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents