Abstract
In the classical covering problems, the goal is to find a subset of vertices/edges that “covers” a specific structure of the graph. In this work, we initiate the study of the covering problems where given a graph G, in addition to the covering, the solution needs to be sparse, i.e., the number of edges with both the endpoints in the solution are minimized. We consider two well-studied covering problems, namely Vertex Cover and Feedback Vertex Set. In Sparse Vertex Cover, given a graph G, and integers k, t, the goal is to find a minimal vertex cover S of size at most k such that the number of edges in G[S] is at most t. Analogously, we can define Sparse Feedback Vertex Set. Both the problems are NP-hard. We studied these problems in the realm of parameterized complexity. Our results are as follows:
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1.
Sparse Vertex Cover admits an \(\mathcal {O}(k^2)\) vertex kernel and an algorithm that runs in \(\mathcal {O}(1.3953^k\cdot n^{O(1)})\) time.
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2.
Sparse Feedback Vertex Set admits an \(\mathcal {O}(k^4)\) vertex kernel and an algorithm that runs in \(\mathcal {O}(5^k\cdot n^{O(1)})\) time.
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Notes
- 1.
The notation \({\mathcal O}^\star (\cdot )\) suppresses the polynomial factor in (n, m).
- 2.
The correctness of Reduction Rules, Lemma, Theorem marked by \(\clubsuit \) are in appendix.
References
Agrawal, A., Gupta, S., Saurabh, S., Sharma, R.: Improved algorithms and combinatorial bounds for independent feedback vertex set. In: Guo, J., Hermelin, D. (eds.) IPEC, pp. 2:1–2:14 (2016)
Agrawal, A., Jain, P., Kanesh, L., Lokshtanov, D., Saurabh, S.: Conflict free feedback vertex set: a parameterized dichotomy. In: Potapov, I., Spirakis, P.G., Worrell, J. (eds.) MFCS, pp. 53:1–53:15 (2018)
Agrawal, A., Jain, P., Kanesh, L., Saurabh, S.: Parameterized complexity of conflict-free matchings and paths. Algorithmica 82(7), 1939–1965 (2020)
Arkin, E.M., et al.: Conflict-free covering. In: CCCG (2015)
Banik, A., Panolan, F., Raman, V., Sahlot, V., Saurabh, S.: Parameterized complexity of geometric covering problems having conflicts. Algorithmica 82(1), 1–19 (2020)
Bonamy, M., Dabrowski, K.K., Feghali, C., Johnson, M., Paulusma, D.: Independent feedback vertex sets for graphs of bounded diameter. Inf. Process. Lett. 131, 26–32 (2018)
Bonamy, M., Dabrowski, K.K., Feghali, C., Johnson, M., Paulusma, D.: Independent feedback vertex set for p 5-free graphs. Algorithmica, pp. 1342–1369 (2019)
Cranston, D.W., Yancey, M.P.: Vertex partitions into an independent set and a forest with each component small. SIAM J. Discret. Math. 35(3), 1769–1791 (2021)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Dabrowski, K.K., Johnson, M., Paesani, G., Paulusma, D., Zamaraev, V.: On the price of independence for vertex cover, feedback vertex set and odd cycle transversal. In: MFCS, pp. 63:1–63:15 (2018)
Diestel, R.: Graph theory. Springer, New York (2000)
Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discret. Math. 313(7), 839–854 (2013)
Goddard, W., Henning, M.A.: Independent domination in outerplanar graphs. Discret. Appl. Math. 325, 52–57 (2023)
Hakimi, S.L., Schmeichel, E.F.: A note on the vertex arboricity of a graph. SIAM J. Discret. Math. 2(1), 64–67 (1989)
Jacob, A., Majumdar, D., Raman, V.: Parameterized complexity of conflict-free set cover. Theory Comput. Syst. 65(3), 515–540 (2021)
Jain, P., Kanesh, L., Misra, P.: Conflict free version of covering problems on graphs: classical and parameterized. Theory Comput. Syst. 64(6), 1067–1093 (2020)
Jain, P., Kanesh, L., Roy, S.K., Saurabh, S., Sharma, R.: Circumventing connectivity for kernelization. In: CIAC 2021, vol. 12701, pp. 300–313 (2021)
Kociumaka, T., Pilipczuk, M.: Faster deterministic feedback vertex set. Inf. Process. Lett. 114(10), 556–560 (2014)
Li, S., Pilipczuk, M.: An improved fpt algorithm for independent feedback vertex set. Theory Comput. Syst. 64, 1317–1330 (2020)
Misra, N., Philip, G., Raman, V., Saurabh, S.: On parameterized independent feedback vertex set. Theor. Comput. Sci. 461, 65–75 (2012)
Thomassé, S.: A 4\({k}{}^{\text{2}}\) kernel for feedback vertex set. ACM Trans. Algorithms 6(2), 32:1–32:8 (2010)
Wu, Y., Yuan, J., Zhao, Y.: Partition a graph into two induced forests. J. Math. Study 1(01) (1996)
Yang, A., Yuan, J.: Partition the vertices of a graph into one independent set and one acyclic set. Discret. Math. 306(12), 1207–1216 (2006)
Yang, A., Yuan, J.: On the vertex arboricity of planar graphs of diameter two. Discret. Math. 307(19–20), 2438–2447 (2007)
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Jain, P., Rathore, M.S. (2024). Sparsity in Covering Solutions. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14579. Springer, Cham. https://doi.org/10.1007/978-3-031-55601-2_9
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