Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T19:45:48.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Assessing the computational complexity of multilayer subgraph detection

Published online by Cambridge University Press:  05 August 2019

Robert Bredereck ,
Christian Komusiewicz ,
Stefan Kratsch ,
Hendrik Molter ,
Rolf Niedermeier  and
Manuel Sorge
Robert Bredereck
Affiliation:
Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany (e-mail: robert.bredereck@tu-berlin.de)
Christian Komusiewicz
Affiliation:
Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany (e-mail: komusiewicz@informatik.uni-marburg.de)
Stefan Kratsch
Affiliation:
Humboldt-Universität zu Berlin, Berlin, Germany (e-mail: kratsch@informatik.hu-berlin.de)
Hendrik Molter*
Affiliation:
Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany (e-mails: h.molter@tu-berlin.de, rolf.niedermeier@tu-berlin.de)
Rolf Niedermeier
Affiliation:
Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany (e-mails: h.molter@tu-berlin.de, rolf.niedermeier@tu-berlin.de)
Manuel Sorge
Affiliation:
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland (e-mail: manuel.sorge@mimuw.edu.pl)
*
*Corresponding author. Email: h.molter@tu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Multilayer graphs consist of several graphs, called layers, where the vertex set of all layers is the same but each layer has an individual edge set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multilayer graphs, including fundamental problems such as maximum-cardinality matching, finding certain clique relaxations, or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of computational tractability.

Keywords

parameterized computational complexityexact algorithmsmulti-modal datamatchingHamiltonian pathcommunity detection
Type
Original Article
Information
Network Science , Volume 7 , Issue 2 , June 2019 , pp. 215 - 241
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

An extended abstract (Bredereck et al., 2017) appeared in the proceedings of the 10th International Conference on Algorithms and Complexity (CIAC 2017), held in Athens, Greece, May 24–26, 2017. This long version now contains a reorganization and much broader motivation and interpretation of the results, as well as full proofs of all results. Work started while all authors were with TU Berlin.

References

Agrawal, A., Krithika, R., Lokshtanov, D., Mouawad, A., & Ramanujan, M. S. (2018a). On the parameterized complexity of simultaneous deletion problems. Proceedings of the 37th IARCS annual conference on foundations of software technology and theoretical computer science (FSTTCS ’17) (pp. 9:1–9:14). Leibniz International Proceedings in Informatics (LIPIcs), vol. 93. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.Google Scholar
Agrawal, A., Lokshtanov, D., Mouawad, A., & Saurabh, S. (2018b). Simultaneous feedback vertex set: A parameterized perspective. ACM Transactions on Computation Theory, 10(4), 18:1–18:25.CrossRefGoogle Scholar
Bang-Jensen, J., & Gutin, G. (2009). Digraphs: theory, algorithms and applications (2nd ed.). Springer.CrossRefGoogle Scholar
Berlingerio, M., Coscia, M., Giannotti, F., Monreale, A., & Pedreschi, D. (2013). Multidimensional networks: foundations of structural analysis. Proceedings of the 22nd International World Wide Web Conference (WWW ’13), 16(5–6), 567–593.CrossRefGoogle Scholar
van Bevern, R. (2014). Towards optimal and expressive kernelization for -hitting set. Algorithmica, 70(1), 129–147.Google Scholar
Boccaletti, S., Bianconi, G., Criado, R., del Genio, C. I., Gómez-Gardeñes, J., Romance, M., … Zanin, M. (2014). The structure and dynamics of multilayer networks. Physics Reports, 544(1), 1–122.CrossRefGoogle Scholar
Boden, B., Günnemann, S., Hoffmann, H., & Seidl, T. (2017). Mimag: mining coherent subgraphs in multi-layer graphs with edge labels. Knowledge and Information Systems, 50(2), 417–446.CrossRefGoogle Scholar
Borgatti, S. P., & Everett, M. G. (2000). Models of core/periphery structures. Social Networks, 21(4), 375–395.CrossRefGoogle Scholar
Brandstädt, A., Le, V. B., & Spinrad, J. P. (1999). Graph classes: A survey. SIAM Monographs on Discrete Mathematics and Applications, vol. 3. SIAM.Google Scholar
Bredereck, R., Komusiewicz, C., Kratsch, S., Molter, H., Niedermeier, R., & Sorge, M. (2017). Assessing the computational complexity of multi-layer subgraph detection. Proceedings of the 10th international conference on algorithms and complexity (CIAC ’17) (vol. 10236, pp. 128–139). LNCS. Springer.Google Scholar
Bui-Xuan, B.-M., Habib, M., & Paul, C. (2008). Competitive graph searches. Theoretical Computer Science, 393(1–3), 72–80.CrossRefGoogle Scholar
Cai, L. (1996). Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58(4), 171–176.CrossRefGoogle Scholar
Cai, L., & Ye, J. (2014). Dual connectedness of edge-bicolored graphs and beyond. Proceedings of the 39th international symposium on mathematical foundations of computer science (MFCS ’14) (vol. 8635., pp. 141–152). LNCS, Springer.Google Scholar
Chen, J., Molter, H., Sorge, M., & Suchý, O. (2018a). Cluster editing in multi-layer and temporal graphs. Proceedings of the 29th international symposium on algorithms and computation (ISAAC ’18) (vol. 123, pp. 24:1–24:13). LIPIcs, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Google Scholar
Chen, J., Niedermeier, R., & Skowron, P. (2018b). Stable marriage with multi-modal preferences. Proceedings of the 2018 ACM conference on economics and computation (EC ’18) (pp. 269–286). ACM.CrossRefGoogle Scholar
Cohen, J. (2008). Trusses: Cohesive subgraphs for social network analysis. National Security Agency Technical Report, 16.Google Scholar
Cygan, M., Fomin, F. V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., … Saurabh, S. (2015). Parameterized algorithms. Springer.CrossRefGoogle Scholar
Dell, H., & Marx, D. (2012). Kernelization of packing problems. Proceedings of the 23rd annual ACM-SIAM symposium on discrete algorithms (SODA ’12) (pp. 68–81). SIAM.CrossRefGoogle Scholar
Diestel, R. (2017). Graph theory (vol. 173, 5th ed.). Graduate Texts in Mathematics. Springer.Google Scholar
Dong, X., Frossard, P., Vandergheynst, P., & Nefedov, N. (2012). Clustering with multi-layer graphs: A spectral perspective. IEEE Transactions on Signal Processing, 60(11), 5820–5831.CrossRefGoogle Scholar
Dong, X., Frossard, P., Vandergheynst, P., & Nefedov, N. (2014). Clustering on multi-layer graphs via subspace analysis on Grassmann manifolds. IEEE Transactions on Signal Processing, 62(4), 905–918.CrossRefGoogle Scholar
Downey, R. G., & Fellows, M. R. (2013). Fundamentals of parameterized complexity. Springer.CrossRefGoogle Scholar
Eppstein, D., & Spiro, E. S. (2012). The -index of a graph and its application to dynamic subgraph statistics. Journal of Graph Algorithms and Applications, 16(2), 543–567.CrossRefGoogle Scholar
Erdös, P., & Rado, R. (1960). Intersection theorems for systems of sets. Journal of the London Mathematical Society, 35(1), 85–90.CrossRefGoogle Scholar
Fellows, M. R., Hermelin, D., Rosamond, F. A., & Vialette, S. (2009). On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1), 53–61.CrossRefGoogle Scholar
Fertin, G., Komusiewicz, C., Mohamed-Babou, H., & Rusu, I. (2015). Finding supported paths in heterogeneous networks. Algorithms, 8(4), 810–831.CrossRefGoogle Scholar
Flum, J., & Grohe, M. (2006). Parameterized complexity theory. Springer.Google Scholar
Gai, A.-T., Habib, M., Paul, C., & Raffinot, M. (2003). Identifying common connected components of graphs. Techical Report, RR-LIRMM-03016, LIRMM, Université de Montpellier II.Google Scholar
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. Freeman.Google Scholar
Golumbic, M. C. (2004). Algorithmic graph theory and perfect graphs (vol. 57, 2nd ed.). Annals of Discrete Mathematics. Elsevier B. V.Google Scholar
Holme, P. (2015). Modern temporal network theory: a colloquium. The European Physical Journal B, 88(9), 234.CrossRefGoogle Scholar
Holme, P., & Saramäki, J. (2012). Temporal networks. Physics Reports, 519(3), 97–125.CrossRefGoogle Scholar
Jiang, D., & Pei, J. (2009). Mining frequent cross-graph quasi-cliques. ACM Transactions on Knowledge Discovery from Data, 2(4).CrossRefGoogle Scholar
Johnson, D. S. (1987). The NP-completeness column: An ongoing guide. Journal of Algorithms, 8(3), 438–448.CrossRefGoogle Scholar
Jukna, Stasys. (2011). Extremal Combinatorics—with applications in computer science. Texts in Theoretical Computer Science. An EATCS Series. Springer.Google Scholar
Kano, M., & Li, X. (2008). Monochromatic and heterochromatic subgraphs in edge-colored graphs—A survey. Graphs and Combinatorics, 24(4), 237–263.CrossRefGoogle Scholar
Khot, S., & Raman, V. (2002). Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science, 289(2), 997–1008.CrossRefGoogle Scholar
Kim, J., & Lee, J. (2015). Community detection in multi-layer graphs: A survey. SIGMOD Record, 44(3), 37–48.CrossRefGoogle Scholar
Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., & Porter, M. A. (2014). Multilayer networks. Journal of Complex Networks, 2(3), 203–271.CrossRefGoogle Scholar
Komusiewicz, C., & Niedermeier, R. (2012). New races in parameterized algorithmics. Proceedings of the 37th international symposium on mathematical foundations of computer science (MFCS ’12) (vol. 7464). LNCS. Springer.Google Scholar
Kratsch, S. (2012). Polynomial kernelizations for MIN f+∏1 and MAX NP. Algorithmica 63(1–2), 532–550.CrossRefGoogle Scholar
Latapy, M., Viard, T., & Magnien, C. (2018). Stream graphs and link streams for the modeling of interactions over time. Social Network Analysis and Mining, 8(1), 61.CrossRefGoogle Scholar
Lewis, J. M., & Yannakakis, M. (1980). The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20(2), 219–230.CrossRefGoogle Scholar
Lin, B. (2015). The parameterized complexity of -biclique. Proceedings of the 26th annual ACM-SIAM symposium on discrete algorithms (SODA ’15) (pp. 605–615). SIAM.CrossRefGoogle Scholar
Magnani, M., & Rossi, L. (2011). The ML-model for multi-layer social networks. Proceedings of the international conference on advances in social networks analysis and mining (ASONAM ’11) (pp. 5–12). IEEE Computer Society.CrossRefGoogle Scholar
Michail, O. (2016). An introduction to temporal graphs: An algorithmic perspective. Internet Mathematics, 12(4), 239–280.CrossRefGoogle Scholar
Monien, B. (1985). How to find long paths efficiently. Annals of Discrete Mathematics, 25, 239–254.Google Scholar
Moser, H. (2009). Finding optimal solutions for covering and matching problems. Ph.D. thesis, Institut für Informatik, Friedrich-Schiller Universität Jena.Google Scholar
Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., & Onnela, J.-P. (2010). Community structure in time-dependent, multiscale, and multiplex networks. Science, 328(5980), 876–878.CrossRefGoogle ScholarPubMed
Nastos, J., & Gao, Y. (2013). Familial groups in social networks. Social Networks, 35(3), 439–450.CrossRefGoogle Scholar
Nichterlein, A. (2014). Degree-constrained editing of small-degree graphs. Ph.d. thesis, TU Berlin.Google Scholar
Niedermeier, R. (2006). Invitation to fixed-parameter algorithms. Oxford University Press.CrossRefGoogle Scholar
Niedermeier, R. (2010). Reflections on multivariate algorithmics and problem parameterization. Proceedings of the 27th international symposium on theoretical aspects of computer science (STACS ’10) (vol. 5, pp. 17–32). Germany: LIPIcs IBFI Dagstuhl.Google Scholar
Plummer, M. D. (2007). Graph factors and factorization: 1985-2003: A survey. Discrete Mathematics, 307(7–8), 791–821.CrossRefGoogle Scholar
Rossi, L., Musolesi, M., & Torsello, A. (2015). On the k-anonymization of time-varying and multi-layer social graphs. Proceedings of the 9th international conference on web and social media (ICWSM ’15) (pp. 377–386). AAAI Press.Google Scholar
Seidman, S. B. (1983). Network structure and minimum degree. Social Networks, 5(3), 269–287.CrossRefGoogle Scholar
Zeng, Z., Wang, J., Zhou, L., & Karypis, G. (2007). Out-of-core coherent closed quasi-clique mining from large dense graph databases. ACM Transactions on Database Systems, 32(2), 13.CrossRefGoogle Scholar