{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T17:25:45Z","timestamp":1761845145910,"version":"3.37.3"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,2,4]],"date-time":"2020-02-04T00:00:00Z","timestamp":1580774400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,4]],"date-time":"2020-02-04T00:00:00Z","timestamp":1580774400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Membr Comput"],"published-print":{"date-parts":[[2020,3]]},"DOI":"10.1007\/s41965-019-00030-1","type":"journal-article","created":{"date-parts":[[2020,2,4]],"date-time":"2020-02-04T11:04:22Z","timestamp":1580814262000},"page":"26-41","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Quantum solutions for densest k-subgraph problems"],"prefix":"10.1007","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8711-6799","authenticated-orcid":false,"given":"Cristian S.","family":"Calude","sequence":"first","affiliation":[]},{"given":"Michael J.","family":"Dinneen","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Hua","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,4]]},"reference":[{"key":"30_CR1","unstructured":"Abbott, A. A., Calude, C. S., Dinneen, M. J., & Hua, R. (2018). A hybrid quantum-classical paradigm to mitigate embedding costs in quantum annealing. CDMTCS Report Series 520."},{"issue":"3","key":"30_CR2","first-page":"171","volume":"43","author":"A Billionnet","year":"2005","unstructured":"Billionnet, A. (2005). Different formulations for solving the heaviest k-subgraph problem. INFOR: Information Systems and Operational Research, 43(3), 171\u2013186.","journal-title":"INFOR: Information Systems and Operational Research"},{"issue":"1","key":"30_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0012-365X(81)90253-3","volume":"33","author":"B Bollob\u00e1s","year":"1981","unstructured":"Bollob\u00e1s, B. (1981). Degree sequences of random graphs. Discrete Mathematics, 33(1), 1\u201319.","journal-title":"Discrete Mathematics"},{"key":"30_CR4","first-page":"1","volume-title":"chap. The maximum clique problem","author":"IM Bomze","year":"1999","unstructured":"Bomze, I. M., Budinich, M., Pardalos, P., & Pelillo, M. (1999). Handbook of combinatorial optimization. In D.-Z. Du & P. M. Pardalos (Eds.), chap. The maximum clique problem (pp. 1\u201374). Dordrecht: Kluwer Academic Publishers."},{"issue":"1","key":"30_CR5","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s11128-015-1150-6","volume":"15","author":"T Boothby","year":"2016","unstructured":"Boothby, T., King, A. D., & Roy, A. (2016). Fast clique minor generation in Chimera quit connectivity graphs. Quantum Information Processing, 15(1), 495\u2013508.","journal-title":"Quantum Information Processing"},{"issue":"1","key":"30_CR6","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1145\/2744447.2744459","volume":"46","author":"CS Calude","year":"2015","unstructured":"Calude, C. S., Calude, E., & Dinneen, M. J. (2015). Adiabatic quantum computing challenges. ACM SIGACT News, 46(1), 40\u201361. https:\/\/doi.org\/10.1145\/2744447.2744459.","journal-title":"ACM SIGACT News"},{"key":"30_CR7","doi-asserted-by":"publisher","unstructured":"Calude, C. S., Dinneen, M. J., & Hua, R. (2017). QUBO formulations for the graph isomorphism problem and related problems. Theoretical Computer Science. https:\/\/doi.org\/10.1016\/j.tcs.2017.04.016.","DOI":"10.1016\/j.tcs.2017.04.016"},{"key":"30_CR8","unstructured":"Calude, C. S., Dinneen, M. J., & Hua, R. (2019). Quantum solutions for densest $$k$$-subgraph problems. Report CDMTCS-540, Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland, Auckland, New Zealand. https:\/\/www.cs.auckland.ac.nz\/research\/groups\/CDMTCS\/researchreports. Accessed Dec 2019."},{"key":"30_CR9","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1186\/s13059-015-0865-0","volume":"17","author":"S Canzar","year":"2016","unstructured":"Canzar, S., Andreotti, S., Weese, D., Reinert, K., & Klau, G. W. (2016). CIDANE: Comprehensive isoform discovery and abundance estimation. Genome Biology, 17, 16.","journal-title":"Genome Biology"},{"issue":"1","key":"30_CR10","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1016\/0166-218X(84)90088-X","volume":"9","author":"DG Corneil","year":"1984","unstructured":"Corneil, D. G., & Perl, Y. (1984). Clustering and domination in perfect graphs. Discrete Applied Mathematics, 9(1), 27\u201339.","journal-title":"Discrete Applied Mathematics"},{"key":"30_CR11","unstructured":"D-Wave Systems. (2018). D-Wave Problem-Solving Handbook. User Manual. https:\/\/docs.dwavesys.com\/docs\/latest\/doc_handbook.html. Accessed Dec 2019."},{"key":"30_CR12","unstructured":"D-Wave Systems. (2018). D-Wave Solver Properties and Parameters Reference. User Manual. https:\/\/docs.dwavesys.com\/docs\/latest\/doc_solver_ref.html. Accessed Dec 2019."},{"key":"30_CR13","unstructured":"D-Wave Systems. (2018). Postprocessing Methods on D-Wave Systems. User Manual. https:\/\/docs.dwavesys.com\/docs\/latest\/doc_post-processing.html. Accessed Dec 2019."},{"key":"30_CR14","unstructured":"D-Wave Systems. (2019). Next-Generation Topology of D-Wave Quantum Processors. https:\/\/www.dwavesys.com\/sites\/default\/files\/14-1026A- C_Next-Generation-Topology-of-DW-Quantum-Processors.pdf. Accessed Dec 2019."},{"key":"30_CR15","unstructured":"D-Wave Systems, Inc. (2018). Developer guide for Python. Technical Report Release 2.4 09-1024A-K."},{"key":"30_CR16","doi-asserted-by":"publisher","unstructured":"Dinneen, M. J., & Hua, R. (2017). Formulating graph covering problems for adiabatic quantum computers. In: Proceedings of the Australasian Computer Science Week Multiconference, ACSW \u201917 (pp. 18:1\u201318:10). New York, NY, USA: ACM. https:\/\/doi.org\/10.1145\/3014812.3014830.","DOI":"10.1145\/3014812.3014830"},{"issue":"14","key":"30_CR17","doi-asserted-by":"publisher","first-page":"e150","DOI":"10.1093\/bioinformatics\/btl243","volume":"22","author":"E Fratkin","year":"2006","unstructured":"Fratkin, E., Naughton, B. T., Brutlag, D. L., & Batzoglou, S. (2006). Motifcut: Regulatory motifs finding with maximum density subgraphs. Bioinformatics, 22(14), e150\u20131e57.","journal-title":"Bioinformatics"},{"key":"30_CR18","unstructured":"Hagberg, A., Schult, D., & Swart, P. (2019). NetworkX. Software for Complex Networks"},{"issue":"5","key":"30_CR19","doi-asserted-by":"publisher","first-page":"eaau0823","DOI":"10.1126\/sciadv.aau0823","volume":"5","author":"R Hamerly","year":"2019","unstructured":"Hamerly, R., Inagaki, T., McMahon, P. L., Venturelli, D., Marandi, A., Onodera, T., et al. (2019). Experimental investigation of performance differences between coherent ising machines and a quantum annealer. Science Advances, 5(5), eaau0823.","journal-title":"Science Advances"},{"issue":"1","key":"30_CR20","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s42979-019-0020-1","volume":"1","author":"R Hua","year":"2020","unstructured":"Hua, R., & Dinneen, M. J. (2020). Improved QUBO formulation of the graph isomorphism problem. SN Computer Science, 1(1), 19.","journal-title":"SN Computer Science"},{"key":"30_CR21","first-page":"269","volume":"4","author":"FV Jensen","year":"1990","unstructured":"Jensen, F. V., Lauritzen, S. L., & Olesen, K. G. (1990). Bayesian updating incausal probabilistic networks by local computations. Computational Statistics Quarterly, 4, 269\u2013282.","journal-title":"Computational Statistics Quarterly"},{"issue":"5","key":"30_CR22","doi-asserted-by":"publisher","first-page":"5355","DOI":"10.1103\/PhysRevE.58.5355","volume":"58","author":"T Kadowaki","year":"1998","unstructured":"Kadowaki, T., & Nishimori, H. (1998). Quantum annealing in the transverse ising model. Physical Review E, 58(5), 5355.","journal-title":"Physical Review E"},{"key":"30_CR23","first-page":"155","volume":"9","author":"JM Keil","year":"1991","unstructured":"Keil, J. M., & Brecht, T. B. (1991). The complexity of clustering in planar graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 9, 155\u2013159.","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"30_CR24","doi-asserted-by":"publisher","unstructured":"Khuller, S., Saha, B. (2009) On finding dense subgraphs. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I, ICALP \u201909 (pp. 597\u2013608). Berlin: Springer. https:\/\/doi.org\/10.1007\/978-3-642-02927-1_50.","DOI":"10.1007\/978-3-642-02927-1_50"},{"issue":"3","key":"30_CR25","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1287\/mnsc.3.3.255","volume":"3","author":"HM Markowitz","year":"1957","unstructured":"Markowitz, H. M. (1957). The elimination form of the inverse and its application to linear programming. Management Science, 3(3), 255\u2013269.","journal-title":"Management Science"},{"key":"30_CR26","doi-asserted-by":"publisher","DOI":"10.2200\/S00585ED1V01Y201407QMC008","volume-title":"Adiabatic quantum computation and quantum annealing. Theory and practice","author":"CC McGeoch","year":"2014","unstructured":"McGeoch, C. C. (2014). Adiabatic quantum computation and quantum annealing. Theory and practice. San Rafael: Morgan & Claypool Publishers."},{"key":"30_CR27","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1109\/MC.2019.2908836","volume":"52","author":"CC McGeoch","year":"2019","unstructured":"McGeoch, C. C., Harris, R., Reinhardt, S. P., & Bunyk, P. I. (2019). Practical annealing-based quantum computing. Computer, 52, 38\u201346.","journal-title":"Computer"},{"key":"30_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-11467-0","volume-title":"The oxford handbook of membrane computing","author":"G Paun","year":"2010","unstructured":"Paun, G., Rozenberg, G., & Salomaa, A. (2010). The oxford handbook of membrane computing. New York: Oxford University Press Inc."},{"key":"30_CR29","doi-asserted-by":"publisher","first-page":"3243","DOI":"10.1038\/ncomms4243","volume":"5","author":"KL Pudenz","year":"2014","unstructured":"Pudenz, K. L., Albash, T., & Lidar, D. A. (2014). Error-corrected quantum annealing with hundreds of qubits. Nature Communications, 5, 3243.","journal-title":"Nature Communications"},{"key":"30_CR30","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1007\/978-3-642-12683-3_30","volume-title":"Research in computational molecular biology","author":"B Saha","year":"2010","unstructured":"Saha, B., Hoch, A., Khuller, S., Raschid, L., & Zhang, X. N. (2010). Dense subgraphs with restrictions and applications to gene annotation graphs. In B. Berger (Ed.), Research in computational molecular biology (pp. 456\u2013472). Berlin: Springer."},{"key":"30_CR31","unstructured":"Stein, W., et\u00a0al. (2016). Sage mathematics software (version 7.0)."}],"container-title":["Journal of Membrane Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s41965-019-00030-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s41965-019-00030-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s41965-019-00030-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,3]],"date-time":"2021-02-03T01:26:08Z","timestamp":1612315568000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s41965-019-00030-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,4]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["30"],"URL":"https:\/\/doi.org\/10.1007\/s41965-019-00030-1","relation":{},"ISSN":["2523-8906","2523-8914"],"issn-type":[{"type":"print","value":"2523-8906"},{"type":"electronic","value":"2523-8914"}],"subject":[],"published":{"date-parts":[[2020,2,4]]},"assertion":[{"value":"11 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 December 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 February 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}