Document Open Access Logo

Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing

Authors Elfarouk Harb, Kent Quanrud, Chandra Chekuri

Thumbnail PDF


  • Filesize: 0.76 MB
  • 17 pages

Document Identifiers

Author Details

Elfarouk Harb
  • University of Illinois at Urbana Champaign, IL, USA
Kent Quanrud
  • Purdue University, West Lafayette, IN, USA
Chandra Chekuri
  • University of Illinois at Urbana Champaign, IL, USA

Cite AsGet BibTex

Elfarouk Harb, Kent Quanrud, and Chandra Chekuri. Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 56:1-56:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Boob et al. [Boob et al., 2020] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Qaunrud and Torres [Chandra Chekuri et al., 2022] extended the algorithm to supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige’s quadratic program for finding a lexicographically optimal base in a (contra) polymatroid [Satoru Fujishige, 1980], and a noisy version of the Frank-Wolfe method from convex optimization [Frank and Wolfe, 1956; Jaggi, 2013]. This yields a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [Harb et al., 2022]. A second contribution of the paper is to understand Thorup’s work on ideal tree packing and greedy tree packing [Thorup, 2007; Thorup, 2008] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige’s result and convex optimization.

Subject Classification

ACM Subject Classification
  • Networks → Network algorithms
  • Mathematics of computing → Graph algorithms
  • Polymatroid
  • lexicographically optimum base
  • densest subgraph
  • tree packing


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Reid Andersen and Kumar Chellapilla. Finding dense subgraphs with size bounds. In Konstantin Avrachenkov, Debora Donato, and Nelly Litvak, editors, Algorithms and Models for the Web-Graph, pages 25-37, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. Google Scholar
  2. Albert Angel, Nikos Sarkas, Nick Koudas, and Divesh Srivastava. Dense subgraph maintenance under streaming edge weight updates for real-time story identification. Proc. VLDB Endow., 5(6):574-585, February 2012. URL:
  3. Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of computing, 8(1):121-164, 2012. Google Scholar
  4. Yuichi Asahiro, Kazuo Iwama, Hisao Tamaki, and Takeshi Tokuyama. Greedily finding a dense subgraph. Journal of Algorithms, 34(2):203-221, 2000. Google Scholar
  5. Bahman Bahmani, Ashish Goel, and Kamesh Munagala. Efficient primal-dual graph algorithms for mapreduce. In International Workshop on Algorithms and Models for the Web-Graph, pages 59-78. Springer, 2014. Google Scholar
  6. Oana Denisa Balalau, Francesco Bonchi, T-H. Hubert Chan, Francesco Gullo, and Mauro Sozio. Finding subgraphs with maximum total density and limited overlap. In Proceedings of the Eighth ACM International Conference on Web Search and Data Mining, WSDM '15, pages 379-388, New York, NY, USA, 2015. Association for Computing Machinery. URL:
  7. Digvijay Boob, Yu Gao, Richard Peng, Saurabh Sawlani, Charalampos Tsourakakis, Di Wang, and Junxing Wang. Flowless: Extracting Densest Subgraphs Without Flow Computations, pages 573-583. Association for Computing Machinery, New York, NY, USA, 2020. URL:
  8. Digvijay Boob, Saurabh Sawlani, and Di Wang. Faster width-dependent algorithm for mixed packing and covering lps. Advances in Neural Information Processing Systems 32 (NIPS 2019), 2019. Google Scholar
  9. Moses Charikar. Greedy approximation algorithms for finding dense components in a graph. In Klaus Jansen and Samir Khuller, editors, Approximation Algorithms for Combinatorial Optimization, pages 84-95, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg. Google Scholar
  10. Chandra Chekuri, Kent Quanrud, and Manuel R. Torres. Densest subgraph: Supermodularity, iterative peeling, and flow. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1531-1555, 2022. URL:
  11. Chandra Chekuri, Kent Quanrud, and Chao Xu. Lp relaxation and tree packing for minimum k-cut. SIAM Journal on Discrete Mathematics, 34(2):1334-1353, 2020. Google Scholar
  12. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time, 2022. URL:
  13. Aleksander B. G. Christiansen, Jacob Holm, Ivor van der Hoog, Eva Rotenberg, and Chris Schwiegelshohn. Adaptive out-orientations with applications, 2023. URL:
  14. Maximilien Danisch, T.-H. Hubert Chan, and Mauro Sozio. Large scale density-friendly graph decomposition via convex programming. In Proceedings of the 26th International Conference on World Wide Web, WWW '17, pages 233-242, Republic and Canton of Geneva, CHE, 2017. International World Wide Web Conferences Steering Committee. URL:
  15. Jack Edmonds. Submodular functions, matroids, and certain polyhedra. In R. Guy, H. Hanani, N. Sauer, and J. Schönheim, editors, Combinatorial Structures and Their Applications (Proceedings Calgary International Conference on Combinatorial Structures and Their Applications, Calgary, Alberta, 1969; R. Guy, H. Hanani, N. Sauer, J. Schönheim, eds.), New York, 1970. Gordon and Breach. Google Scholar
  16. Alessandro Epasto, Silvio Lattanzi, and Mauro Sozio. Efficient densest subgraph computation in evolving graphs. In Proceedings of the 24th International Conference on World Wide Web, WWW '15, pages 300-310, Republic and Canton of Geneva, CHE, 2015. International World Wide Web Conferences Steering Committee. URL:
  17. Marguerite Frank and Philip Wolfe. An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3(1-2):95-110, 1956. URL:
  18. Satoru Fujishige. Lexicographically optimal base of a polymatroid with respect to a weight vector. Mathematics of Operations Research, 5(2):186-196, 1980. URL:
  19. Satoru Fujishige. Submodular functions and optimization. Elsevier, 2005. Google Scholar
  20. Satoru Fujishige. Theory of principal partitions revisited. Research Trends in Combinatorial Optimization: Bonn 2008, pages 127-162, 2009. Google Scholar
  21. Takuro Fukunaga. Computing minimum multiway cuts in hypergraphs from hypertree packings. In IPCO, pages 15-28. Springer, 2010. Google Scholar
  22. A. V. Goldberg. Finding a maximum density subgraph. Technical Report UCB/CSD-84-171, EECS Department, University of California, Berkeley, 1984. URL:
  23. Anupam Gupta, David G Harris, Euiwoong Lee, and Jason Li. Optimal bounds for the k-cut problem. ACM Journal of the ACM (JACM), 69(1):1-18, 2021. Google Scholar
  24. Elfarouk Harb, Kent Quanrud, and Chandra Chekuri. Faster and scalable algorithms for densest subgraph and decomposition. In Advances in Neural Information Processing Systems, volume 35, pages 26966-26979. Curran Associates, Inc., 2022. URL:
  25. Martin Jaggi. Revisiting Frank-Wolfe: Projection-free sparse convex optimization. In Sanjoy Dasgupta and David McAllester, editors, Proceedings of the 30th International Conference on Machine Learning, number 1 in Proceedings of Machine Learning Research, pages 427-435, Atlanta, Georgia, USA, 17-19 June 2013. PMLR. URL:
  26. David R Karger. Minimum cuts in near-linear time. Journal of the ACM (JACM), 47(1):46-76, 2000. Google Scholar
  27. Yuko Kuroki, Atsushi Miyauchi, Junya Honda, and Masashi Sugiyama. Online dense subgraph discovery via blurred-graph feedback. In ICML, 2020. Google Scholar
  28. Tommaso Lanciano, Atsushi Miyauchi, Adriano Fazzone, and Francesco Bonchi. A survey on the densest subgraph problem and its variants, 2023. URL:
  29. Jason Li. Faster minimum k-cut of a simple graph. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1056-1077. IEEE, 2019. Google Scholar
  30. Xiangfeng Li, Shenghua Liu, Zifeng Li, Xiaotian Han, Chuan Shi, Bryan Hooi, He Huang, and Xueqi Cheng. Flowscope: Spotting money laundering based on graphs. In AAAI, 2020. Google Scholar
  31. Daniel Lokshtanov, Saket Saurabh, and Vaishali Surianarayanan. A parameterized approximation scheme for min k-cut. SIAM Journal on Computing, 0:FOCS20-205, 2022. Google Scholar
  32. Chenhao Ma, Yixiang Fang, Reynold Cheng, Laks V.S. Lakshmanan, Wenjie Zhang, and Xuemin Lin. Efficient algorithms for densest subgraph discovery on large directed graphs. In Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data, SIGMOD '20, pages 1051-1066, New York, NY, USA, 2020. Association for Computing Machinery. URL:
  33. Fragkiskos D Malliaros, Christos Giatsidis, Apostolos N Papadopoulos, and Michalis Vazirgiannis. The core decomposition of networks: Theory, algorithms and applications. The VLDB Journal, 29(1):61-92, 2020. Google Scholar
  34. Andrew McGregor, David Tench, Sofya Vorotnikova, and Hoa T. Vu. Densest subgraph in dynamic graph streams. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald T. Sannella, editors, Mathematical Foundations of Computer Science 2015, pages 472-482, Berlin, Heidelberg, 2015. Springer Berlin Heidelberg. Google Scholar
  35. H Narayanan. The principal lattice of partitions of a submodular function. Linear Algebra and its Applications, 144:179-216, 1991. Google Scholar
  36. Hariharan Narayanan. Submodular functions and electrical networks, volume 54. Elsevier, 1997. Google Scholar
  37. Jean-Claude Picard and Maurice Queyranne. A network flow solution to some nonlinear 0-1 programming problems, with applications to graph theory. Networks, 12(2):141-159, 1982. Google Scholar
  38. Serge A. Plotkin, David B. Shmoys, and Éva Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20(2):257-301, 1995. URL:
  39. Polina Rozenshtein, Nikolaj Tatti, and Aristides Gionis. Discovering dynamic communities in interaction networks. In Toon Calders, Floriana Esposito, Eyke Hüllermeier, and Rosa Meo, editors, Machine Learning and Knowledge Discovery in Databases, pages 678-693, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. Google Scholar
  40. Saurabh Sawlani and Junxing Wang. Near-optimal fully dynamic densest subgraph. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 181-193. ACM, 2020. URL:
  41. Alexander Schrijver et al. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  42. Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos. Corescope: Graph mining using k-core analysis — patterns, anomalies and algorithms. In 2016 IEEE 16th International Conference on Data Mining (ICDM), pages 469-478, 2016. URL:
  43. Bintao Sun, Maximilien Danisch, T-H. Hubert Chan, and Mauro Sozio. Kclist++: A simple algorithm for finding k-clique densest subgraphs in large graphs. Proc. VLDB Endow., 13(10):1628-1640, June 2020. URL:
  44. Nikolaj Tatti. Density-friendly graph decomposition. ACM Transactions on Knowledge Discovery from Data (TKDD), 13(5):1-29, 2019. Google Scholar
  45. Nikolaj Tatti and Aristides Gionis. Density-friendly graph decomposition. In Proceedings of the 24th International Conference on World Wide Web, pages 1089-1099, 2015. Google Scholar
  46. Mikkel Thorup. Fully-dynamic min-cut. Combinatorica, 27(1):91-127, 2007. Preliminary version in Proc. of ACM STOC 2001. Google Scholar
  47. Mikkel Thorup. Minimum k-way cuts via deterministic greedy tree packing. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 159-166, 2008. Google Scholar
  48. Charalampos Tsourakakis. The k-clique densest subgraph problem. In Proceedings of the 24th International Conference on World Wide Web, WWW '15, pages 1122-1132, Republic and Canton of Geneva, CHE, 2015. International World Wide Web Conferences Steering Committee. URL:
  49. Charalampos Tsourakakis, Francesco Bonchi, Aristides Gionis, Francesco Gullo, and Maria Tsiarli. Denser than the densest subgraph: Extracting optimal quasi-cliques with quality guarantees. In Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '13, pages 104-112, New York, NY, USA, 2013. Association for Computing Machinery. URL:
  50. N. Young. Randomized rounding without solving the linear program. In ACM-SIAM Symposium on Discrete Algorithms, 1995. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail