Document

# Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs

## File

LIPIcs.MFCS.2024.15.pdf
• Filesize: 0.82 MB
• 18 pages

## Cite As

Tian Bai and Mingyu Xiao. Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.15

## Abstract

The Subset Feedback Vertex Set problem (SFVS) is to delete k vertices from a given graph such that in the remaining graph, any vertex in a subset T of vertices (called a terminal set) is not in a cycle. The famous Feedback Vertex Set problem is the special case of SFVS with T being the whole set of vertices. In this paper, we study exact algorithms for SFVS in Split Graphs (SFVS-S) and SFVS in Chordal Graphs (SFVS-C). SFVS-S generalizes the minimum vertex cover problem and the prize-collecting version of the maximum independent set problem in hypergraphs (PCMIS), and SFVS-C further generalizes SFVS-S. Both SFVS-S and SFVS-C are implicit 3-Hitting Set problems. However, it is not easy to solve them faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in 𝒪^*(2^k) time, slightly improving the best result 𝒪^*(2.0755^k) for 3-Hitting Set. In this paper, we break the "2^k-barrier" for SFVS-S and SFVS-C by introducing an 𝒪^*(1.8192^k)-time algorithm. This achievement also indicates that PCMIS can be solved in 𝒪^*(1.8192ⁿ) time, marking the first exact algorithm for PCMIS that outperforms the trivial 𝒪^*(2ⁿ) threshold. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Subset Feedback Vertex Set
• Prize-Collecting Maximum Independent Set
• Parameterized Algorithms
• Split Graphs
• Chordal Graphs
• Dulmage-Mendelsohn Decomposition

## Metrics

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

## References

1. Faisal N. Abu-Khzam. A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci., 76(7):524-531, 2010.
2. Tian Bai and Mingyu Xiao. Exact and parameterized algorithms for restricted subset feedback vertex set in chordal graphs. In Theory and Applications of Models of Computation - 17th Annual Conference, TAMC, volume 13571 of LNCS, pages 249-261. Springer, 2022.
3. Tian Bai and Mingyu Xiao. A parameterized algorithm for subset feedback vertex set in tournaments. Theor. Comput. Sci., 975:114139, 2023.
4. Tian Bai and Mingyu Xiao. Exact algorithms for restricted subset feedback vertex set in chordal and split graphs. Theor. Comput. Sci., 984:114326, 2024. URL: https://doi.org/10.1016/J.TCS.2023.114326.
5. Jannis Blauth and Martin Nägele. An improved approximation guarantee for prize-collecting TSP. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, (STOC), pages 1848-1861. ACM, 2023.
6. Hans L. Bodlaender. On disjoint cycles. Int. J. Found. Comput. Sci., 5(1):59-68, 1994.
7. Peter Buneman. A characterisation of rigid circuit graphs. Discrete Mathematics, 9(3):205-212, 1974.
8. Chandra Chekuri and Chao Xu. Computing minimum cuts in hypergraphs. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA), pages 1085-1100. SIAM, 2017.
9. Jianer Chen and Iyad A. Kanj. Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithms. J. Comput. Syst. Sci., 67(4):833-847, 2003.
10. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, Berlin, 2015.
11. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Subset feedback vertex set is fixed-parameter tractable. SIAM J. Discret. Math., 27(1):290-309, 2013.
12. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dániel Marx. 09511 open problems – parameterized complexity and approximation algorithms. In Parameterized complexity and approximation algorithms, volume 9511 of Dagstuhl Seminar Proceedings (DagSemProc), pages 1-10, Dagstuhl, Germany, 2010.
13. Gabriel Andrew Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25(1):71-76, 1961.
14. Michael Dom, Jiong Guo, Falk Hüffner, Rolf Niedermeier, and Anke Truß. Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms, 8(1):76-86, 2010.
15. A. L. Dulmage and N. S. Mendelsohn. Coverings of bipartite graphs. Canadian Journal of Mathematics, 10:517-534, 1958.
16. Andrew L Dulmage. A structure theory of bipartite graphs of finite exterior dimension. The Transactions of the Royal Society of Canada, Section III, 53:1-13, 1959.
17. Guy Even, Joseph Naor, and Leonid Zosin. An 8-approximation algorithm for the subset feedback vertex set problem. SIAM J. Comput., 30(4):1231-1252, 2000.
18. Fedor V. Fomin, Pinar Heggernes, Dieter Kratsch, Charis Papadopoulos, and Yngve Villanger. Enumerating minimal subset feedback vertex sets. Algorithmica, 69(1):216-231, 2014.
19. Fedor V. Fomin, Tien-Nam Le, Daniel Lokshtanov, Saket Saurabh, Stéphan Thomassé, and Meirav Zehavi. Subquadratic kernels for implicit 3-hitting set and 3-set packing problems. ACM Trans. Algorithms, 15(1):13:1-13:44, 2019.
20. Kyle Fox, Debmalya Panigrahi, and Fred Zhang. Minimum cut and minimum k-cut in hypergraphs via branching contractions. ACM Trans. Algorithms, 19(2):13:1-13:22, 2023.
21. Takuro Fukunaga. Spider covers for prize-collecting network activation problem. ACM Trans. Algorithms, 13(4):49:1-49:31, 2017.
22. Delbert Fulkerson and Oliver Gross. Incidence matrices and interval graphs. Pacific J. Math., 15(3):835-855, 1965.
23. Fǎnicǎ Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974.
24. Magnús M. Halldórsson and Elena Losievskaja. Independent sets in bounded-degree hypergraphs. Discret. Appl. Math., 157(8):1773-1786, 2009.
25. Eva-Maria C. Hols and Stefan Kratsch. A randomized polynomial kernel for subset feedback vertex set. Theory Comput. Syst., 62(1):63-92, 2018.
26. John E. Hopcroft and Richard M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput., 2(4):225-231, 1973.
27. Falk Hüffner, Christian Komusiewicz, Hannes Moser, and Rolf Niedermeier. Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst., 47(1):196-217, 2010.
28. Yoichi Iwata. Linear-time kernelization for feedback vertex set. In 44th International Colloquium on Automata, Languages, and Programming, ICALP, pages 68:1-68:14, 2017.
29. Yoichi Iwata and Yusuke Kobayashi. Improved analysis of highest-degree branching for feedback vertex set. Algorithmica, 83(8):2503-2520, 2021.
30. Yoichi Iwata, Magnus Wahlström, and Yuichi Yoshida. Half-integrality, LP-branching, and FPT algorithms. SIAM J. Comput., 45(4):1377-1411, 2016.
31. Yoichi Iwata, Yutaro Yamaguchi, and Yuichi Yoshida. 0/1/all CSPs, half-integral A-path packing, and linear-time FPT algorithms. In 59th IEEE Annual Symposium on Foundations of Computer Science, (FOCS), pages 462-473, 2018.
32. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a Symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103, 1972.
33. Mithilesh Kumar and Daniel Lokshtanov. Faster exact and parameterized algorithm for feedback vertex set in tournaments. In 33rd Symposium on Theoretical Aspects of Computer Science, (STACS), volume 47 of LIPIcs, pages 49:1-49:13, 2016.
34. László Lovász and Michael D. Plummer. Matching theory, volume 121 of North-Holland Mathematics Studies. Elsevier Science Ltd., London, 1 edition, 1986.
35. Charis Papadopoulos and Spyridon Tzimas. Polynomial-time algorithms for the subset feedback vertex set problem on interval graphs and permutation graphs. Discret. Appl. Math., 258:204-221, 2019.
36. Lehilton Lelis Chaves Pedrosa and Hugo Kooki Kasuya Rosado. A 2-approximation for the k-prize-collecting steiner tree problem. Algorithmica, 84(12):3522-3558, 2022.
37. Geevarghese Philip, Varun Rajan, Saket Saurabh, and Prafullkumar Tale. Subset feedback vertex set in chordal and split graphs. Algorithmica, 81(9):3586-3629, 2019.
38. Donald J. Rose, Robert Endre Tarjan, and George S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput., 5(2):266-283, 1976.
39. Földes Stephane and Peter Hammer. Split graphs. In Proceedings of the 8th south-east Combinatorics, Graph Theory, and Computing, volume 9, pages 311-315, 1977.
40. Stéphan Thomassé. A 4k² kernel for feedback vertex set. ACM Trans. Algorithms, 6(2):32:1-32:8, 2010.
41. Kangyi Tian, Mingyu Xiao, and Boting Yang. Parameterized algorithms for cluster vertex deletion on degree-4 graphs and general graphs. In Computing and Combinatorics - 29th International Conference, (COCOON), volume 14422 of LNCS, pages 182-194. Springer, 2023.
42. Magnus Wahlström. Algorithms, measures and upper bounds for satisfiability and related problems. PhD thesis, Linköping University, Sweden, 2007.
43. James Richard Walter. Representations of rigid cycle graphs. PhD thesis, Wayne State University, 1972.
44. Mihalis Yannakakis and Fanica Gavril. The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett., 24(2):133-137, 1987.
X

Feedback for Dagstuhl Publishing