Abstract
In this paper, we investigate the complexity of the Maximum Happy Set problem on subclasses of co-comparability graphs. For a graph G and its vertex subset S, a vertex \(v \in S\) is happy if all v’s neighbors in G are contained in S. Given a graph G and a non-negative integer k, Maximum Happy Set is the problem of finding a vertex subset S of G such that \(|S|= k\) and the number of happy vertices in S is maximized. In this paper, we first show that Maximum Happy Set is NP-hard even for co-bipartite graphs. We then give an algorithm for n-vertex interval graphs whose running time is \(O(n^2 + k^3n)\); this improves the best known running time \(O(kn^8)\) for interval graphs. We also design algorithms for n-vertex permutation graphs and d-trapezoid graphs which run in \(O(n^2 + k^3n)\) and \(O(n^2 + d^2(k+1)^{3d}n)\) time, respectively. These algorithmic results provide a nice contrast to the fact that Maximum Happy Set remains NP-hard for chordal graphs, comparability graphs, and co-comparability graphs.
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Notes
We note that the graph coloring problem introduced by Zhang and Li [3] is called a similar name, Maximum Happy Vertices, but it is a different problem from ours.
References
Eto, H., Ito, T., Miyano, E., Suzuki, A., Tamura, Y.: Happy set problem on subclasses of co-comparability graphs. In: Mutzel, P., Rahman, M.S., Slamin (eds.) Proceedings of the 16th International Conference and Workshop on Algorithms and Computation (WALCOM 2022). Lecture Notes in Computer Science, vol. 13174, pp. 149–160. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-96731-4_13
Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511761942
Zhang, P., Li, A.: Algorithmic aspects of homophyly of networks. Theor. Comput. Sci. 593, 117–131 (2015). https://doi.org/10.1016/j.tcs.2015.06.003
Asahiro, Y., Eto, H., Hanaka, T., Lin, G., Miyano, E., Terabaru, I.: Parameterized algorithms for the happy set problem. Discrete Appl. Math. 304, 32–44 (2021). https://doi.org/10.1016/j.dam.2021.07.005
Asahiro, Y., Eto, H., Hanaka, T., Lin, G., Miyano, E., Terabaru, I.: Complexity and approximability of the happy set problem. Theor. Comput. Sci. 866, 123–144 (2021). https://doi.org/10.1016/j.tcs.2021.03.023
Kratsch, D., Stewart, L.: Domination on cocomparability graphs. SIAM J. Discrete Math. 6(3), 400–417 (1993). https://doi.org/10.1137/0406032
Deogun, J.S., Steiner, G.: Polynomial algorithms for Hamiltonian cycle in cocomparability graphs. SIAM J. Comput. 23(3), 520–552 (1994). https://doi.org/10.1137/S0097539791200375
Coorg, S.R., Rangan, C.P.: Feedback vertex set on cocomparability graphs. Networks 26(2), 101–111 (1995). https://doi.org/10.1002/net.3230260205
Feige, U., Kortsarz, G., Peleg, D.: The dense \(k\)-subgraph problem. Algorithmica 29(3), 410–421 (2001). https://doi.org/10.1007/s004530010050
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976). https://doi.org/10.1016/S0022-0000(76)80045-1
Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discrete Math. 23(4), 1905–1953 (2010). https://doi.org/10.1137/S0895480100373455
Hsu, W.-L., Ma, T.-H.: Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs. SIAM J. Comput. 28(3), 1004–1020 (1998). https://doi.org/10.1137/S0097539792224814
McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Math. 201(1), 189–241 (1999). https://doi.org/10.1016/S0012-365X(98)00319-7
Bodlaender, H.L., Kloks, T., Kratsch, D., Müller, H.: Treewidth and minimum fill-in on \(d\)-trapezoid graphs. J. Graph Algorithms Appl. 2(5), 1–23 (1998). https://doi.org/10.7155/jgaa.00008
Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebr. Discrete Methods 3(3), 351–358 (1982). https://doi.org/10.1137/0603036
Ma, T.-H., Spinrad, J.P.: On the 2-chain subgraph cover and related problems. J. Algorithms 17(2), 251–268 (1994). https://doi.org/10.1006/jagm.1994.1034
Acknowledgements
We are grateful to anonymous referees of the preliminary version [1] and of this journal version for their helpful suggestions. This work is partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP19K11814, JP20H05793, JP20H05794, JP20K11666, JP21K11755, and JP21K21278, Japan.
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Eto, H., Ito, T., Miyano, E. et al. Happy Set Problem on Subclasses of Co-comparability Graphs. Algorithmica 85, 3327–3347 (2023). https://doi.org/10.1007/s00453-022-01081-0
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DOI: https://doi.org/10.1007/s00453-022-01081-0