A Survey on the Densest Subgraph Problem and its Variants

The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature over the last five decades, with many variants proposed and many applications built on top of this basic definition. Recent years have witnessed a revival of research interest on this problem with several interesting contributions, including some groundbreaking results, published in 2022 and 2023. This survey provides a deep overview of the fundamental results and an exhaustive coverage of the many variants proposed in the literature, with a special attention on the most recent results. The survey also presents a comprehensive overview of applications and discusses some interesting open problems for this evergreen research topic.

[1]  Zhuo Tang,et al.  Scalable Algorithms for Densest Subgraph Discovery , 2023, 2023 IEEE 39th International Conference on Data Engineering (ICDE).

[2]  Arijit Khan,et al.  Most Probable Densest Subgraphs , 2022, 2023 IEEE 39th International Conference on Data Engineering (ICDE).

[3]  Johan Kok Zhi Kang,et al.  Spade: A Real-Time Fraud Detection Framework on Evolving Graphs (Complete Version) , 2022, Proc. VLDB Endow..

[4]  Yasushi Kawase,et al.  Stochastic Solutions for Dense Subgraph Discovery in Multilayer Networks , 2022, WSDM.

[5]  T. Akutsu,et al.  Densest subgraph-based methods for protein-protein interaction hot spot prediction , 2022, BMC Bioinformatics.

[6]  L. Becchetti,et al.  Network Based Approach to Gene Prioritization at Genome-Wide Association Study Loci , 2022, 2210.16292.

[7]  C. Chekuri,et al.  $(1-\epsilon)$-approximate fully dynamic densest subgraph: linear space and faster update time , 2022, 2210.02611.

[8]  Quanquan C. Liu,et al.  Differential Privacy from Locally Adjustable Graph Algorithms: k-Core Decomposition, Low Out-Degree Ordering, and Densest Subgraphs , 2022, 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS).

[9]  Pingpeng Yuan,et al.  Discovering Cohesive Temporal Subgraphs with Temporal Density Aware Exploration , 2022, Journal of Computer Science and Technology.

[10]  E. Rotenberg,et al.  Adaptive Out-Orientations with Applications , 2022, ArXiv.

[11]  Charalampos E. Tsourakakis,et al.  Discovering Polarization Niches via Dense Subgraphs with Attractors and Repulsers , 2022, Proc. VLDB Endow..

[12]  M. Henzinger,et al.  Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs , 2022, ESA.

[13]  Zheng-Chao Zhang,et al.  Detecting Cash-out Users via Dense Subgraphs , 2022, KDD.

[14]  P. Provero,et al.  Contrast subgraphs allow comparing homogeneous and heterogeneous networks derived from omics data , 2022, bioRxiv.

[15]  T. Hanaka.  Computing densest k-subgraph with structural parameters , 2022, Journal of Combinatorial Optimization.

[16]  Yi Zhou,et al.  Extracting Densest Sub-hypergraph with Convex Edge-weight Functions , 2022, TAMC.

[17]  Yinghui Wu,et al.  A Stochastic Approach to Finding Densest Temporal Subgraphs in Dynamic Graphs , 2022, IEEE Transactions on Knowledge and Data Engineering.

[18]  N. Sidiropoulos,et al.  The Triangle-Densest-K-Subgraph Problem: Hardness, Lovász Extension, and Application to Document Summarization , 2022, AAAI.

[19]  Atsushi Miyauchi,et al.  A Study on Modularity Density Maximization: Column Generation Acceleration and Computational Complexity Analysis , 2022, Eur. J. Oper. Res..

[20]  L. Lakshmanan,et al.  A Convex-Programming Approach for Efficient Directed Densest Subgraph Discovery , 2022, SIGMOD Conference.

[21]  Lijun Chang,et al.  Anchored Densest Subgraph , 2022, SIGMOD Conference.

[22]  Charalampos E. Tsourakakis,et al.  AntiBenford Subgraphs: Unsupervised Anomaly Detection in Financial Networks , 2022, KDD.

[23]  Guoren Wang,et al.  Colorful h-star Core Decomposition , 2022, 2022 IEEE 38th International Conference on Data Engineering (ICDE).

[24]  Y. Zhang,et al.  Hierarchical Core Decomposition in Parallel: From Construction to Subgraph Search , 2022, 2022 IEEE 38th International Conference on Data Engineering (ICDE).

[25]  L. Lakshmanan,et al.  FirmCore Decomposition of Multilayer Networks , 2022, WWW.

[26]  Suman Kalyan Bera,et al.  A New Dynamic Algorithm for Densest Subhypergraphs , 2022, WWW.

[27]  Manuel R. Torres,et al.  Densest Subgraph: Supermodularity, Iterative Peeling, and Flow , 2022, SODA.

[28]  Charalampos E. Tsourakakis,et al.  Dense and well-connected subgraph detection in dual networks , 2021, SDM.

[29]  Laks V. S. Lakshmanan,et al.  On Directed Densest Subgraph Discovery , 2021, ACM Trans. Database Syst..

[30]  The Wedge Picking Model: A Theoretical Analysis of Graph Evolution Caused by Triadic Closure and Algorithmic Implications , 2021, Journal of Strategic Innovation and Sustainability.

[31]  Deke Guo,et al.  Anomaly Detection of Network Streams via Dense Subgraph Discovery , 2021, 2021 International Conference on Computer Communications and Networks (ICCCN).

[32]  Reynold Cheng,et al.  Efficient Directed Densest Subgraph Discovery , 2021, SIGMOD Rec..

[33]  Austin R. Benson,et al.  The Generalized Mean Densest Subgraph Problem , 2021, KDD.

[34]  E. Shi,et al.  Differentially Private Densest Subgraph , 2021, AISTATS.

[35]  Anil Vullikanti,et al.  Differentially Private Densest Subgraph Detection , 2021, ICML.

[36]  Jianliang Xu,et al.  Efficient Probabilistic Truss Indexing on Uncertain Graphs , 2021, WWW.

[37]  Riccardo Dondi,et al.  Dense Sub-networks Discovery in Temporal Networks , 2021, SN Computer Science.

[38]  Sanghamitra Bandyopadhyay,et al.  Supervised feature selection using integration of densest subgraph finding with floating forward-backward search , 2021, Inf. Sci..

[39]  N. Sidiropoulos,et al.  Exploring the Subgraph Density-Size Trade-off via the Lovaśz Extension , 2021, WSDM.

[40]  R. Klasing,et al.  The Hardness and Approximation of the Densest k-Subgraph Problem in Parameterized Metric Graphs , 2020, 2020 International Computer Symposium (ICS).

[41]  J. Buhmann,et al.  Statistical and computational thresholds for the planted k-densest sub-hypergraph problem , 2020, AISTATS.

[42]  Jonathan P. How,et al.  CLIPPER: A Graph-Theoretic Framework for Robust Data Association , 2020, 2021 IEEE International Conference on Robotics and Automation (ICRA).

[43]  Luca Becchetti,et al.  Spectral Relaxations and Fair Densest Subgraphs , 2020, CIKM.

[44]  J. Waltz,et al.  Extracting Brain Disease-Related Connectome Subgraphs by Adaptive Dense Subgraph Discovery , 2020, bioRxiv.

[45]  Zhendong Liu,et al.  Approximating Max k-Uncut via LP-rounding Plus Greed, with Applications to Densest k-Subgraph , 2020, AAIM.

[46]  Aristides Gionis,et al.  Mining Dense Subgraphs with Similar Edges , 2020, ECML/PKDD.

[47]  Charalampos E. Tsourakakis,et al.  Finding Densest k-Connected Subgraphs , 2020, Discret. Appl. Math..

[48]  Minghui Zhou,et al.  Scaling Open Source Communities: An Empirical Study of the Linux Kernel , 2020, 2020 IEEE/ACM 42nd International Conference on Software Engineering (ICSE).

[49]  Masashi Sugiyama,et al.  Online Dense Subgraph Discovery via Blurred-Graph Feedback , 2020, ICML.

[50]  Jingyu Hou,et al.  Prediction optimization of diffusion paths in social networks using integration of ant colony and densest subgraph algorithms , 2020, J. High Speed Networks.

[51]  Gao Cong,et al.  Densely Connected User Community and Location Cluster Search in Location-Based Social Networks , 2020, SIGMOD Conference.

[52]  Aristides Gionis,et al.  Explainable Classification of Brain Networks via Contrast Subgraphs , 2020, KDD.

[53]  Anand Louis,et al.  Planted Models for the Densest $k$-Subgraph Problem , 2020, FSTTCS.

[54]  Bryan Hooi,et al.  FlowScope: Spotting Money Laundering Based on Graphs , 2020, AAAI.

[55]  Shuai Ma,et al.  An Efficient Approach to Finding Dense Temporal Subgraphs , 2020, IEEE Transactions on Knowledge and Data Engineering.

[56]  Julian Shun,et al.  Parallel Clique Counting and Peeling Algorithms , 2020, ACDA.

[57]  Vahab Mirrokni,et al.  Contextual Reserve Price Optimization in Auctions , 2020, NeurIPS.

[58]  Georgios A. Pavlopoulos,et al.  A Guide to Conquer the Biological Network Era Using Graph Theory , 2020, Frontiers in Bioengineering and Biotechnology.

[59]  Liefeng Bo,et al.  EnsemFDet: An Ensemble Approach to Fraud Detection based on Bipartite Graph , 2019, 2021 IEEE 37th International Conference on Data Engineering (ICDE).

[60]  Michele Monaci,et al.  In search of dense subgraphs: How good is greedy peeling? , 2019, Networks.

[61]  Michalis Vazirgiannis,et al.  The core decomposition of networks: theory, algorithms and applications , 2019, The VLDB Journal.

[62]  Richard Peng,et al.  Flowless: Extracting Densest Subgraphs Without Flow Computations , 2019, WWW.

[63]  Di Wang,et al.  Faster width-dependent algorithm for mixed packing and covering LPs , 2019, NeurIPS.

[64]  Jian Pei,et al.  Online Density Bursting Subgraph Detection from Temporal Graphs , 2019, Proc. VLDB Endow..

[65]  András Faragó,et al.  In Search of the Densest Subgraph , 2019, Algorithms.

[66]  Hsin-Hao Su,et al.  Distributed Dense Subgraph Detection and Low Outdegree Orientation , 2019, DISC.

[67]  Theresa Migler,et al.  The Densest k Subgraph Problem in b-Outerplanar Graphs , 2019, COMPLEX NETWORKS.

[68]  Saurabh Sawlani,et al.  Near-optimal fully dynamic densest subgraph , 2019, STOC.

[69]  Reynold Cheng,et al.  Efficient Algorithms for Densest Subgraph Discovery , 2019, Proc. VLDB Endow..

[70]  Silvio Lattanzi,et al.  Improved Parallel Algorithms for Density-Based Network Clustering , 2019, ICML.

[71]  Seny Kamara,et al.  Computationally Volume-Hiding Structured Encryption , 2019, EUROCRYPT.

[72]  Yasushi Kawase,et al.  Graph Mining Meets Crowdsourcing: Extracting Experts for Answer Aggregation , 2019, IJCAI.

[73]  Jakub W. Pachocki,et al.  Novel Dense Subgraph Discovery Primitives: Risk Aversion and Exclusion Queries , 2019, ECML/PKDD.

[74]  Francesco Bonchi,et al.  Distance-generalized Core Decomposition , 2019, SIGMOD Conference.

[75]  Brendan Ames,et al.  Convex Optimization for the Densest Subgraph and Densest Submatrix Problems , 2019, SN Operations Research Forum.

[76]  Renata Sotirov,et al.  On solving the densest k-subgraph problem on large graphs , 2019, Optim. Methods Softw..

[77]  Lijun Chang,et al.  Cohesive Subgraph Computation Over Large Sparse Graphs: Algorithms, Data Structures, and Programming Techniques , 2018 .

[78]  Francesco Bonchi,et al.  Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications. , 2018, 1812.08712.

[79]  Naonori Kakimura,et al.  Finding a Dense Subgraph with Sparse Cut , 2018, CIKM.

[80]  Xin Liu,et al.  No Place to Hide: Catching Fraudulent Entities in Tensors , 2018, WWW.

[81]  Akiko Takeda,et al.  Robust Densest Subgraph Discovery , 2018, 2018 IEEE International Conference on Data Mining (ICDM).

[82]  Aristides Gionis,et al.  Finding events in temporal networks: segmentation meets densest subgraph discovery , 2018, Knowledge and Information Systems.

[83]  Michael Mitzenmacher,et al.  Metric Sublinear Algorithms via Linear Sampling , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[84]  Moses Charikar,et al.  On Finding Dense Common Subgraphs , 2018, ArXiv.

[85]  Enhong Chen,et al.  Mining Density Contrast Subgraphs , 2018, 2018 IEEE 34th International Conference on Data Engineering (ICDE).

[86]  Charalampos E. Tsourakakis,et al.  Risk-Averse Matchings over Uncertain Graph Databases , 2018, ECML/PKDD.

[87]  Satoshi Hara,et al.  Discounted average degree density metric and new algorithms for the densest subgraph problem , 2018, Networks.

[88]  Francesco Bonchi,et al.  Core Decomposition and Densest Subgraph in Multilayer Networks , 2017, CIKM.

[89]  T.-H. Hubert Chan,et al.  Maintaining Densest Subsets Efficiently in Evolving Hypergraphs , 2017, CIKM.

[90]  Ümit V. Çatalyürek,et al.  Nucleus Decompositions for Identifying Hierarchy of Dense Subgraphs , 2017, ACM Trans. Web.

[91]  Aristides Gionis,et al.  Finding Dynamic Dense Subgraphs , 2017, ACM Trans. Knowl. Discov. Data.

[92]  Yasushi Kawase,et al.  The Densest Subgraph Problem with a Convex/Concave Size Function , 2017, Algorithmica.

[93]  Evaggelia Pitoura,et al.  Finding lasting dense subgraphs , 2016, Data Mining and Knowledge Discovery.

[94]  Charu C. Aggarwal,et al.  On Dense Subgraphs in Signed Network Streams , 2016, 2016 IEEE 16th International Conference on Data Mining (ICDM).

[95]  Pasin Manurangsi,et al.  Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph , 2016, STOC.

[96]  Erik Norlander,et al.  Finding the Densest Common Subgraph with Linear Programming , 2016 .

[97]  Francesco Bonchi,et al.  Identifying Buzzing Stories via Anomalous Temporal Subgraph Discovery , 2016, 2016 IEEE/WIC/ACM International Conference on Web Intelligence (WI).

[98]  Hyun Ah Song,et al.  FRAUDAR: Bounding Graph Fraud in the Face of Camouflage , 2016, KDD.

[99]  Michael Dinitz,et al.  The Densest k-Subhypergraph Problem , 2016, APPROX-RANDOM.

[100]  Frédéric Roupin,et al.  Computational results of a semidefinite branch-and-bound algorithm for k-cluster , 2016, Comput. Oper. Res..

[101]  Niko Beerenwinkel,et al.  Finding Dense Subgraphs in Relational Graphs , 2015, ECML/PKDD.

[102]  Yin Tat Lee,et al.  A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[103]  Charalampos E. Tsourakakis,et al.  Dense Subgraph Discovery: KDD 2015 tutorial , 2015, KDD.

[104]  Jakub W. Pachocki,et al.  Scalable Large Near-Clique Detection in Large-Scale Networks via Sampling , 2015, KDD.

[105]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[106]  Takuro Fukunaga,et al.  Threshold Influence Model for Allocating Advertising Budgets , 2015, ICML.

[107]  David P. Woodruff,et al.  Brief Announcement: Applications of Uniform Sampling: Densest Subgraph and Beyond , 2015, SPAA.

[108]  Sofya Vorotnikova,et al.  Densest Subgraph in Dynamic Graph Streams , 2015, MFCS.

[109]  Charalampos E. Tsourakakis.  The K-clique Densest Subgraph Problem , 2015, WWW.

[110]  Silvio Lattanzi,et al.  Efficient Densest Subgraph Computation in Evolving Graphs , 2015, WWW.

[111]  Qinghua Wu,et al.  A review on algorithms for maximum clique problems , 2015, Eur. J. Oper. Res..

[112]  Omri Weinstein,et al.  ETH Hardness for Densest-k-Subgraph with Perfect Completeness , 2015, SODA.

[113]  Xiaofeng Zhu,et al.  Finding dense and connected subgraphs in dual networks , 2015, 2015 IEEE 31st International Conference on Data Engineering.

[114]  Charalampos E. Tsourakakis,et al.  Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams , 2015, STOC.

[115]  Francesco Bonchi,et al.  Finding Subgraphs with Maximum Total Density and Limited Overlap , 2015, WSDM.

[116]  Kamesh Munagala,et al.  Efficient Primal-Dual Graph Algorithms for MapReduce , 2014, WAW.

[117]  Wei Chen,et al.  Combinatorial Pure Exploration of Multi-Armed Bandits , 2014, NIPS.

[118]  Ümit V. Çatalyürek,et al.  Finding the Hierarchy of Dense Subgraphs using Nucleus Decompositions , 2014, WWW.

[119]  Aristides Gionis,et al.  Event detection in activity networks , 2014, KDD.

[120]  Francesco Bonchi,et al.  Core decomposition of uncertain graphs , 2014, KDD.

[121]  Dimitris S. Papailiopoulos,et al.  Finding Dense Subgraphs via Low-Rank Bilinear Optimization , 2014, ICML.

[122]  Siddharth Barman,et al.  Approximating Nash Equilibria and Dense Subgraphs via an Approximate Version of Carathéodory's Theorem , 2014, SIAM J. Comput..

[123]  Roberto Navigli,et al.  Entity Linking meets Word Sense Disambiguation: a Unified Approach , 2014, TACL.

[124]  Peter Sanders,et al.  Recent Advances in Graph Partitioning , 2013, Algorithm Engineering.

[125]  Charalampos E. Tsourakakis,et al.  Denser than the densest subgraph: extracting optimal quasi-cliques with quality guarantees , 2013, KDD.

[126]  Michalis Vazirgiannis,et al.  Clustering and Community Detection in Directed Networks: A Survey , 2013, ArXiv.

[127]  Matthias Hein,et al.  Towards realistic team formation in social networks based on densest subgraphs , 2013, WWW.

[128]  Vangelis Th. Paschos,et al.  Exact and Approximation Algorithms for Densest k-Subgraph , 2013, WALCOM.

[129]  Wei Chen,et al.  Combinatorial Multi-Armed Bandit: General Framework and Applications , 2013, ICML.

[130]  Frédéric Roupin,et al.  Solving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-cluster problems to optimality with semidefi , 2012, Mathematical Programming.

[131]  Avrim Blum,et al.  Differentially private data analysis of social networks via restricted sensitivity , 2012, ITCS '13.

[132]  Ashwin Lall,et al.  Dense Subgraphs on Dynamic Networks , 2012, DISC.

[133]  Sanjeev Arora,et al.  The Multiplicative Weights Update Method: a Meta-Algorithm and Applications , 2012, Theory Comput..

[134]  Divesh Srivastava,et al.  Dense subgraph maintenance under streaming edge weight updates for real-time story identification , 2012, The VLDB Journal.

[135]  Sergei Vassilvitskii,et al.  Densest Subgraph in Streaming and MapReduce , 2012, Proc. VLDB Endow..

[136]  Ambuj K. Singh,et al.  Mining Heavy Subgraphs in Time-Evolving Networks , 2011, 2011 IEEE 11th International Conference on Data Mining.

[137]  Petr A. Golovach,et al.  Tight Complexity Bounds for FPT Subgraph Problems Parameterized by Clique-Width , 2011, IPEC.

[138]  Kazuyuki Aihara,et al.  Size-constrained Submodular Minimization through Minimum Norm Base , 2011, ICML.

[139]  André Carlos Ponce de Leon Ferreira de Carvalho,et al.  Spectral methods for graph clustering - A survey , 2011, Eur. J. Oper. Res..

[140]  Tao Jiang,et al.  A max-flow based approach to the identification of protein complexes using protein interaction and microarray data. , 2008, Computational systems bioinformatics. Computational Systems Bioinformatics Conference.

[141]  Sergei Vassilvitskii,et al.  Counting triangles and the curse of the last reducer , 2011, WWW.

[142]  Rudolf Fleischer,et al.  Densest k-Subgraph Approximation on Intersection Graphs , 2010, WAOA.

[143]  George Kollios,et al.  k-nearest neighbors in uncertain graphs , 2010, Proc. VLDB Endow..

[144]  Aristides Gionis,et al.  The community-search problem and how to plan a successful cocktail party , 2010, KDD.

[145]  J. Mark Keil,et al.  Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs , 2010, Inf. Process. Lett..

[146]  R. Munos,et al.  Best Arm Identification in Multi-Armed Bandits , 2010, COLT 2010.

[147]  Qin Zhang,et al.  Optimal sampling from distributed streams , 2010, PODS '10.

[148]  Prasad Raghavendra,et al.  Graph expansion and the unique games conjecture , 2010, STOC '10.

[149]  Samir Khuller,et al.  Dense Subgraphs with Restrictions and Applications to Gene Annotation Graphs , 2010, RECOMB.

[150]  Aditya Bhaskara,et al.  Detecting high log-densities: an O(n¼) approximation for densest k-subgraph , 2010, STOC '10.

[151]  Samir Khuller,et al.  On Finding Dense Subgraphs , 2009, ICALP.

[152]  Yang Xiang,et al.  3-HOP: a high-compression indexing scheme for reachability query , 2009, SIGMOD Conference.

[153]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[154]  Alain Billionnet,et al.  Improving the performance of standard solvers for quadratic 0-1 programs by a tight convex reformulation: The QCR method , 2009, Discret. Appl. Math..

[155]  Kumar Chellapilla,et al.  Finding Dense Subgraphs with Size Bounds , 2009, WAW.

[156]  Robert Krauthgamer,et al.  How hard is it to approximate the best Nash equilibrium? , 2009, SODA.

[157]  Ioannis Milis,et al.  A constant approximation algorithm for the densest k , 2008, Inf. Process. Lett..

[158]  Dorit S. Hochbaum,et al.  The Pseudoflow Algorithm: A New Algorithm for the Maximum-Flow Problem , 2008, Oper. Res..

[159]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

[160]  James B. Orlin,et al.  A faster strongly polynomial time algorithm for submodular function minimization , 2007, Math. Program..

[161]  Serafim Batzoglou,et al.  MotifCut: regulatory motifs finding with maximum density subgraphs , 2006, ISMB.

[162]  Sridhar Hannenhalli,et al.  Dense subgraph computation via stochastic search: application to detect transcriptional modules , 2006, ISMB.

[163]  Subhash Khot,et al.  Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[164]  U. Feige.  Relations between average case complexity and approximation complexity , 2002, STOC '02.

[165]  Edith Cohen,et al.  Reachability and distance queries via 2-hop labels , 2002, SODA '02.

[166]  Michael Langberg,et al.  Approximation Algorithms for Maximization Problems Arising in Graph Partitioning , 2001, J. Algorithms.

[167]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[168]  Uriel Feige,et al.  The Dense k -Subgraph Problem , 2001, Algorithmica.

[169]  Moses Charikar,et al.  Greedy approximation algorithms for finding dense components in a graph , 2000, APPROX.

[170]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[171]  S. Sitharama Iyengar,et al.  Introduction to parallel algorithms , 1998, Wiley series on parallel and distributed computing.

[172]  Refael Hassin,et al.  Approximation algorithms for maximum dispersion , 1997, Oper. Res. Lett..

[173]  Hisao Tamaki,et al.  Greedily Finding a Dense Subgraph , 1996, J. Algorithms.

[174]  Marek Karpinski,et al.  Polynomial time approximation schemes for dense instances of NP-hard problems , 1995, STOC '95.

[175]  Andrew B. Kahng,et al.  When clusters meet partitions: new density-based methods for circuit decomposition , 1995, Proceedings the European Design and Test Conference. ED&TC 1995.

[176]  S. S. Ravi,et al.  Heuristic and Special Case Algorithms for Dispersion Problems , 1994, Oper. Res..

[177]  Guy Kortsarz,et al.  On choosing a dense subgraph , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[178]  Guy Kortsarz,et al.  Generating Sparse 2-Spanners , 1992, J. Algorithms.

[179]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

[180]  Maurice Queyranne,et al.  A network flow solution to some nonlinear 0-1 programming problems, with applications to graph theory , 1982, Networks.

[181]  W. Mader.  Existenzn-fach zusammenhängender Teilgraphen in Graphen genügend großer Kantendichte , 1972 .

[182]  Guoren Wang,et al.  Densest Periodic Subgraph Mining on Large Temporal Graphs , 2023, IEEE Transactions on Knowledge and Data Engineering.

[183]  C. Chekuri,et al.  Faster and Scalable Algorithms for Densest Subgraph and Decomposition , 2022, NeurIPS.

[184]  Blair D. Sullivan,et al.  Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph , 2022, IPEC.

[185]  Charalampos E. Tsourakakis,et al.  Algorithmic Tools for Understanding the Motif Structure of Networks , 2022, ECML/PKDD.

[186]  Yixiang Fang,et al.  Densest Subgraph Discovery on Large Graphs: Applications, Challenges, and Techniques , 2022, Proc. VLDB Endow..

[187]  Xuemin Lin,et al.  Cohesive Subgraph Search Over Large Heterogeneous Information Networks , 2022, Springer Briefs in Computer Science.

[188]  D. Song,et al.  Orion: Zero Knowledge Proof with Linear Prover Time , 2022, IACR Cryptol. ePrint Arch..

[189]  Chenhao Ma,et al.  Finding Locally Densest Subgraphs: A Convex Programming Approach , 2022, Proc. VLDB Endow..

[190]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[191]  C. R. Ramakrishnan,et al.  Efficient Distribution of Quantum Circuits , 2021, DISC.

[192]  Xueqi Cheng,et al.  SpecGreedy: Unified Dense Subgraph Detection , 2020, ECML/PKDD.

[193]  Christian Komusiewicz,et al.  FixCon: A Generic Solver for Fixed-Cardinality Subgraph Problems , 2020, ALENEX.

[194]  Francesco Bonchi,et al.  The importance of unexpectedness: Discovering buzzing stories in anomalous temporal graphs , 2019, Web Intell..

[195]  Satu Elisa Schaeffer,et al.  Graph Clustering , 2017, Encyclopedia of Machine Learning and Data Mining.

[196]  Anthony Wirth,et al.  Correlation Clustering , 2017, Encyclopedia of Machine Learning and Data Mining.

[197]  Zhaonian Zou,et al.  Polynomial-Time Algorithm for Finding Densest Subgraphs in Uncertain Graphs , 2013 .

[198]  Charu C. Aggarwal,et al.  A Survey of Algorithms for Dense Subgraph Discovery , 2010, Managing and Mining Graph Data.

[199]  Leizhen Cai,et al.  Parameterized Complexity of Cardinality Constrained Optimization Problems , 2008, Comput. J..

[200]  Jiawei Han,et al.  Mining coherent dense subgraphs across massive biological networks for functional discovery , 2005, ISMB.

[201]  Jiawei Zhang,et al.  Approximation of Dense-n/2-Subgraph and the Complement of Min-Bisection , 2003, J. Glob. Optim..

[202]  U. Feige,et al.  On the Densest K-subgraph Problem , 1997 .

[203]  藤重 悟.  Submodular functions and optimization , 1991 .

[204]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[205]  Andrew V. Goldberg,et al.  Finding a Maximum Density Subgraph , 1984 .

[206]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..