Abstract
Let G be a graph where every vertex has a colour and has specified diversity constraints, that is, a minimum number of neighbours of every colour. Every vertex also has a max-degree constraint: an upper bound on the total number of neighbours. In the Min-Edit-Cost problem, we wish to transform G using edge additions and/or deletions into a graph \(G'\) where every vertex satisfies all diversity as well as max-degree constraints. We show an \(O(n^5 \log n)\) algorithm for the Min-Edit-Cost problem, and an \(O(n^3 \log n \log \log n)\) algorithm for the bipartite case. Given a specified number of edge operations, the Max-Satisfied-Nodes problem is to find the maximum number of vertices whose diversity constraints can be satisfied while ensuring that all max-degree constraints are satisfied. We show that the Max-Satisfied-Nodes problem is W[1]-hard, in parameter \(r+ \ell \), where r is the number of edge operations and \(\ell \) is the number of vertices to be satisfied. We also show that it is inapproximable to within a factor of \(n^{1/2-\epsilon }\). For certain relaxations of the max-degree constraints, we are able to show constant-factor approximation algorithms for the problem.
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References
Ahuja, R.K., Goldberg, A.V., Orlin, J.B., Tarjan, R.E.: Finding minimum-cost flows by double scaling. Math. Program. 53(1–3), 243–266 (1992)
Bredereck, R., Hartung, S., Nichterlein, A., Woeginger, G.J.: The complexity of finding a large subgraph under anonymity constraints. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 152–162. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-45030-3_15
Cheah, F., Corneil, D.G.: The complexity of regular subgraph recognition. Discrete Appl. Math. 27(1–2), 59–68 (1990)
Chvátal, V., Fleischner, H., Sheehan, J., Thomassen, C.: Three-regular subgraphs of four-regular graphs. J. Graph Theory 3(4), 371–386 (1979)
Cornuéjols, G.: General factors of graphs. J. Comb. Theory Ser. B 45(2), 185–198 (1988)
Dondi, R., Lafond, M., El-Mabrouk, N.: Approximating the correction of weighted and unweighted orthology and paralogy relations. Algorithms Mol. Biol. 12(1), 4 (2017)
Duan, R., Pettie, S., Su, H.-H.: Scaling algorithms for weighted matching in general graphs. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 781–800. Society for Industrial and Applied Mathematics (2017)
Fischer, A., Suen, C.Y., Frinken, V., Riesen, K., Bunke, H.: A fast matching algorithm for graph-based handwriting recognition. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 194–203. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38221-5_21
Gao, X., Xiao, B., Tao, D., Li, X.: A survey of graph edit distance. Pattern Anal. Appl. 13(1), 113–129 (2010)
Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 29. W. H. freeman, New York (2002)
Golovach, P.A.: Editing to a connected graph of given degrees. Inf. Comput. 256, 131–147 (2017)
Golovach, P.A., Mertzios, G.B.: Graph editing to a given degree sequence. Theoret. Comput. Sci. 665, 1–12 (2017)
Hartung, S., Nichterlein, A., Niedermeier, R., Suchỳ, O.: A refined complexity analysis of degree anonymization in graphs. Inf. Comput. 243, 249–262 (2015)
Hu, H., Yan, X., Huang, Y., Han, J., Zhou, X.J.: Mining coherent dense subgraphs across massive biological networks for functional discovery. Bioinformatics 21, i213–i221 (2005)
Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)
Lin, B.: The parameterized complexity of k-Biclique. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 605–615. Society for Industrial and Applied Mathematics (2015)
Liu, K., Terzi, E.: Towards identity anonymization on graphs. In: Proceedings of the 2008 ACM SIGMOD International Conference on Management of Data, pp. 93–106. ACM (2008)
Liu, Y., Wang, J., Guo, J., Chen, J.: Cograph editing: complexity and parameterized algorithms. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 110–121. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22685-4_10
Lovász, L.: The factorization of graphs. ii. Acta Mathematica Academiae Scientiarum Hungarica, 23(1–2), 223–246 (1972)
Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: a parameterized approach. J. Comput. Syst. Sci. 78(1), 179–191 (2012)
Neuhaus, M., Bunke, H.: A graph matching based approach to fingerprint classification using directional variance. In: Kanade, T., Jain, A., Ratha, N.K. (eds.) AVBPA 2005. LNCS, vol. 3546, pp. 191–200. Springer, Heidelberg (2005). https://doi.org/10.1007/11527923_20
Neuhaus, M., Bunke, H.: Bridging the Gap Between Graph Edit Distance and Kernel Machines, vol. 68. World Scientific, Singapore (2007)
Riesen, K., Bunke, H.: Approximate graph edit distance computation by means of bipartite graph matching. Image Vis. Comput. 27(7), 950–959 (2009)
Riesen, K., Neuhaus, M., Bunke, H.: Bipartite graph matching for computing the edit distance of graphs. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 1–12. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72903-7_1
Subramanya, V.: Graph editing to a given neighbourhood degree list is fixed-parameter tractable. Master’s thesis, University of Waterloo (2016)
Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10(2), 297–309 (1981)
Zeng, Z., Tung, A.K.H., Wang, J., Feng, J., Zhou, L.: Comparing stars: on approximating graph edit distance. Proc. VLDB Endow. 2(1), 25–36 (2009)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 681–690. ACM (2006)
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We thank Jaroslav Opatrny for useful discussions.
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Chuangpishit, H., Lafond, M., Narayanan, L. (2018). Editing Graphs to Satisfy Diversity Requirements. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_11
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