Algorithmica

, Volume 81, Issue 9, pp 3586–3629

# Subset Feedback Vertex Set in Chordal and Split Graphs

• Geevarghese Philip
• Varun Rajan
• Saket Saurabh
• Prafullkumar Tale
Article

## Abstract

In the Subset Feedback Vertex Set (Subset-FVS) problem the input is a graph G on n vertices, a subset T of vertices of G called the “terminal” vertices, and an integer k. The task is to determine whether there exists a subset of vertices of cardinality at most k which together intersect all cycles which pass through the terminals. Subset-FVS generalizes several well studied problems including Feedback Vertex Set and Multiway Cut. This problem is known to be NP-Complete, even in split graphs. Cygan et al. (SIAM J Discrete Math 27(1):290–309, 2013) proved that Subset-FVS is fixed parameter tractable ($$\mathsf {FPT}$$) in general graphs when parameterized by k. In split graphs a simple observation reduces the problem to an equivalent instance of the 3-Hitting Set problem with the same solution size. This directly implies, for Subset-FVSrestricted to split graphs, (i) an $$\mathsf {FPT}$$ algorithm which solves the problem in $$\mathcal {O}^{\star } (2.076^k)$$ time (The $$\mathcal {O}^{\star } ()$$ notation hides polynomial factors.) (Wahlström in Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. Thesis, Department of Computer and Information Science, Linköpings universitet, 2007), and (ii) a kernel of size $$\mathcal {O}(k^3)$$. We improve both these results for Subset-FVS on split graphs; we derive (i) a kernel of size $$\mathcal {O}(k^2)$$ which is the best possible unless $$\textsf {NP}\subseteq {\mathsf {coNP}}/{\textsf {poly}}$$, and (ii) an algorithm which solves the problem in time $$\mathcal {O}^*(2^k)$$. Our algorithm, in fact, solves Subset-FVS on the more general class of chordal graphs, also in $$\mathcal {O}^*(2^k)$$ time. To the best of our knowledge, the fastest known exact algorithm for Subset-FVS on chordal graphs is based on the 3-Hitting Set algorithm of Fomin et al. (JACM 66(2):8, 2019) which runs in $$\mathcal {O}^*(1.5182^n)$$ time. Applying the results of Fomin et al. (2019) to our $$\mathsf {FPT}$$ algorithm yields two exact exponential-time algorithms for Subset-FVS on chordal graphs: a randomized algorithm which runs in $$\mathcal {O}^*(1.5^{n})$$ time, and a deterministic algorithm which runs in $$\mathcal {O}^*((1.5+\varepsilon )^{n})$$ time for any fixed $$\varepsilon >0$$.

## Keywords

Subset feedback vertex set Chordal and split graphs Parameterized complexity

## References

1. 1.
Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010).
2. 2.
Blair, J.R., Peyton, B.: An introduction to chordal graphs and clique trees. In: Graph Theory and Sparse Matrix Computation, pp. 1–29. Springer (1993)Google Scholar
3. 3.
Chitnis, R., Fomin, F.V., Lokshtanov, D., Misra, P., Ramanujan, M., Saurabh, S.: Faster exact algorithms for some terminal set problems. J. Comput. Syst. Sci. 88, 195–207 (2017)
4. 4.
Chitnis, R.H., Cygan, M., Hajiaghayi, M.T., Marx, D.: Directed subset feedback vertex set is fixed-parameter tractable. ACM Trans. Algorithms (TALG) 11(4), 28:1–28:28 (2015)
5. 5.
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)
6. 6.
Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Subset feedback vertex set is fixed-parameter tractable. SIAM J. Discrete Math. 27(1), 290–309 (2013)
7. 7.
Dell, H., Van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61(4), 23:1–23:27 (2014)
8. 8.
Diestel, R.: Graph Theory, 5th edn. Springer, Berlin (2016)
9. 9.
Even, G., Naor, J., Zosin, L.: An 8-approximation algorithm for the subset feedback vertex set problem. SIAM J. Comput. 30(4), 1231–1252 (2000)
10. 10.
Fomin, F., Kratsch, D.: Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin Heidelberg (2010)Google Scholar
11. 11.
Fomin, F.V., Gaspers, S., Lokshtanov, D., Saurabh, S.: Exact algorithms via monotone local search. JACM 66(2), 8 (2019)
12. 12.
Fomin, F.V., Heggernes, P., Kratsch, D., Papadopoulos, C., Villanger, Y.: Enumerating minimal subset feedback vertex sets. Algorithmica 69(1), 216–231 (2014)
13. 13.
Fomin, F.V., Le, T.N., Lokshtanov, D., Saurabh, S., Thomassé, S., Zehavi, M.: Subquadratic kernels for implicit 3-hitting set and 3-set packing problems. ACM Trans. Algorithms (TALG) 15(1), 13 (2019)
14. 14.
Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: approximation and kernelization. SIAM J. Discrete Math. 30(1), 383–410 (2016)
15. 15.
Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, Cambridge (2019)
16. 16.
Galinier, P., Habib, M., Paul, C.: Chordal graphs and their clique graphs. In: International Workshop on Graph-Theoretic Concepts in Computer Science, pp. 358–371. Springer (1995)Google Scholar
17. 17.
Golovach, P.A., Heggernes, P., Kratsch, D., Saei, R.: Subset feedback vertex sets in chordal graphs. J. Discrete Algorithms 26, 7–15 (2014)
18. 18.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004)
19. 19.
Hammer, P.L., Simeone, B.: The splittance of a graph. Combinatorica 1(3), 275–284 (1981)
20. 20.
Hols, E.M.C., Kratsch, S.: A randomized polynomial kernel for subset feedback vertex set. Theory Comput. Syst. 62(1), 63–92 (2018)
21. 21.
Kawarabayashi, Ki, Kobayashi, Y.: Fixed-parameter tractability for the subset feedback set problem and the s-cycle packing problem. J. Comb. Theory Ser. B 102(4), 1020–1034 (2012)
22. 22.
Lokshtanov, D., Ramanujan, M., Saurabh, S.: Linear time parameterized algorithms for subset feedback vertex set. ACM Trans. Algorithms (TALG) 14(1), 7 (2018)
23. 23.
Thomassé, S.: A $$4k^{{2}}$$ kernel for feedback vertex set. ACM Trans. Algorithms (TALG) 6(2), 32:1–32:8 (2010)
24. 24.
Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. Thesis, Department of Computer and Information Science, Linköpings universitet (2007)Google Scholar
25. 25.
Wahlström, M.: Half-integrality, LP-branching and FPT algorithms. In: Proceedings of the Twenty-fifth Annual ACM-SIAM symposium on Discrete algorithms, pp. 1762–1781. SIAM (2014)Google Scholar

## Authors and Affiliations

• Geevarghese Philip
• 1
• Varun Rajan
• 2
• Saket Saurabh
• 3
• 4
• Prafullkumar Tale
• 5
Email author
1. 1.Chennai Mathematical Institute and UMI ReLaXChennaiIndia
2. 2.Chennai Mathematical InstituteChennaiIndia
3. 3.The Institute Of Mathematical Sciences, HBNI and UMI ReLaXChennaiIndia
4. 4.Department of InformaticsUniversity of BergenBergenNorway
5. 5.The Institute Of Mathematical Sciences, HBNIChennaiIndia