Document Open Access Logo

Independent Sets in Elimination Graphs with a Submodular Objective

Authors Chandra Chekuri, Kent Quanrud

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2023.24.pdf
  • Filesize: 0.8 MB
  • 22 pages

Document Identifiers

Author Details

Chandra Chekuri
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Kent Quanrud
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA

Funding

  • Chekuri, Chandra: Supported in part by NSF grants CCF-1910149 and CCF-1907937.
  • Quanrud, Kent: Supported in part by NSF grant CCF-2129816.

Cite AsGet BibTex

Chandra Chekuri and Kent Quanrud. Independent Sets in Elimination Graphs with a Submodular Objective. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 24:1-24:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.24

Abstract

Maximum weight independent set (MWIS) admits a 1/k-approximation in inductively k-independent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)-approximation in k-perfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize k-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others [Ye and Borodin, 2012; Kammer and Tholey, 2014]. We consider a generalization of MWIS to a submodular objective. Given a graph G = (V,E) and a non-negative submodular function f: 2^V → ℝ_+, the goal is to approximately solve max_{S ∈ ℐ_G} f(S) where ℐ_G is the set of independent sets of G. We obtain an Ω(1/k)-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least 1/e(k+1). This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively k-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Submodular optimization and polymatroids
Keywords
  • elimination graphs
  • independent set
  • submodular maximization
  • primal-dual

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Shipra Agrawal, Yichuan Ding, Amin Saberi, and Yinyu Ye. Correlation robust stochastic optimization. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 1087-1096. SIAM, 2010. Google Scholar
  2. Karhan Akcoglu, James Aspnes, Bhaskar DasGupta, and Ming-Yang Kao. Opportunity-cost algorithms for combinatorial auctions. In E. J. Kontoghiorghes, B. Rustem, and S. Siokos, editors, Applied Optimization 74: Computational Methods in Decision-Making, Economics and Finance, pages 455-479. Kluwer Academic Publishers, 2002. Google Scholar
  3. A. Badanidiyuru, B. Mirzasoleiman, A. Karbasi, and A. Krause. Streaming submodular optimization: Massive data summarization on the fly. In Proc. 20th ACM Conf. Knowl. Disc. and Data Mining (KDD), pages 671-680, 2014. Google Scholar
  4. Eric Balkanski and Yaron Singer. The adaptive complexity of maximizing a submodular function. In Proceedings of the 50th annual ACM SIGACT symposium on theory of computing, pages 1138-1151, 2018. Google Scholar
  5. Reuven Bar-Yehuda, Magnús M Halldórsson, Joseph Naor, Hadas Shachnai, and Irina Shapira. Scheduling split intervals. SIAM Journal on Computing, 36(1):1-15, 2006. Google Scholar
  6. Jeff A. Bilmes. Submodularity in machine learning and artificial intelligence. CoRR, abs/2202.00132, 2022. URL: https://arxiv.org/abs/2202.00132.
  7. Simon Bruggmann and Rico Zenklusen. Submodular maximization through the lens of linear programming. Mathematics of Operations Research, 44(4):1221-1244, 2019. Google Scholar
  8. N. Buchbinder, M. Feldman, J. Naor, and R. Schwartz. Submodular maximization with cardinality constraints. In Proc. 25th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 1433-1452, 2014. Google Scholar
  9. Niv Buchbinder and Moran Feldman. Deterministic algorithms for submodular maximization problems. ACM Transactions on Algorithms (TALG), 14(3):1-20, 2018. Google Scholar
  10. Niv Buchbinder and Moran Feldman. Submodular functions maximization problems. In Handbook of Approximation Algorithms and Metaheuristics, Second Edition, pages 753-788. Chapman and Hall/CRC, 2018. Google Scholar
  11. Niv Buchbinder, Moran Feldman, and Mohit Garg. Deterministic (1/2+ ε)-approximation for submodular maximization over a matroid. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 241-254. SIAM, 2019. Google Scholar
  12. Niv Buchbinder, Moran Feldman, and Roy Schwartz. Online submodular maximization with preemption. ACM Transactions on Algorithms (TALG), 15(3):1-31, 2019. Google Scholar
  13. G. Calinescu, C. Chekuri, M. Pál, and J. Vondrák. Maximizing a submodular set function subject to a matroid constraint (extended abstract). In Proc. 12th Int. Conf. Int. Prog. Comb. Opt. (IPCO), pages 182-196, 2007. Google Scholar
  14. Gruia Calinescu, Chandra Chekuri, Martin Pal, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011. Google Scholar
  15. Amit Chakrabarti and Sagar Kale. Submodular maximization meets streaming: matchings, matroids, and more. Mathematical Programming, 154:225-247, 2015. Google Scholar
  16. Timothy M Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. Google Scholar
  17. Chandra Chekuri, Shalmoli Gupta, and Kent Quanrud. Streaming algorithms for submodular function maximization. In Automata, Languages, and Programming: 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I 42, pages 318-330. Springer, 2015. Longer version: URL: http://arxiv.org/abs/1504.08024.
  18. Chandra Chekuri, T. S. Jayram, and Jan Vondrák. On multiplicative weight updates for concave and submodular function maximization. In Tim Roughgarden, editor, Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 201-210. ACM, 2015. URL: https://doi.org/10.1145/2688073.2688086.
  19. Chandra Chekuri and Vasilis Livanos. On submodular prophet inequalities and correlation gap. arXiv preprint arXiv:2107.03662, 2021. Google Scholar
  20. Chandra Chekuri and Kent Quanrud. Submodular function maximization in parallel via the multilinear relaxation. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 303-322. SIAM, 2019. Google Scholar
  21. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Submodular function maximization via the multilinear relaxation and contention resolution schemes. SIAM Journal on Computing, 43(6):1831-1879, 2014. Google Scholar
  22. Andrew Clark, Basel Alomair, Linda Bushnell, and Radha Poovendran. Submodularity in dynamics and control of networked systems. Springer, 2016. Google Scholar
  23. Alina Ene, Huy L Nguyen, and Adrian Vladu. Submodular maximization with matroid and packing constraints in parallel. In Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pages 90-101, 2019. Google Scholar
  24. Moran Feldman. Maximization problems with submodular objective functions. PhD thesis, Computer Science Department, Technion, 2013. Google Scholar
  25. Moran Feldman, Amin Karbasi, and Ehsan Kazemi. Do less, get more: Streaming submodular maximization with subsampling. Advances in Neural Information Processing Systems, 31, 2018. Google Scholar
  26. Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 570-579. IEEE, 2011. Google Scholar
  27. Moran Feldman, Joseph Naor, Roy Schwartz, and Justin Ward. Improved approximations for k-exchange systems. In Algorithms-ESA 2011: 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings 19, pages 784-798. Springer, 2011. Google Scholar
  28. M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey. An analysis of approximations for maximizing submodular set functions - II. Math. Prog. Studies, 8:73-87, 1978. Google Scholar
  29. Paritosh Garg, Linus Jordan, and Ola Svensson. Semi-streaming algorithms for submodular matroid intersection. Mathematical Programming, pages 1-24, 2022. Google Scholar
  30. Magnús M. Halldórsson and Tigran Tonoyan. Computing inductive vertex orderings. Information Processing Letters, 172:106159, 2021. URL: https://doi.org/10.1016/j.ipl.2021.106159.
  31. Kai Han, zongmai Cao, Shuang Cui, and Benwei Wu. Deterministic approximation for submodular maximization over a matroid in nearly linear time. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 430-441. Curran Associates, Inc., 2020. URL: https://proceedings.neurips.cc/paper/2020/file/05128e44e27c36bdba71221bfccf735d-Paper.pdf.
  32. Johan Håstad. Clique is hard to approximate within n^1-ε. Acta Math, 182, 1999. Google Scholar
  33. Th Jenkyns. The efficacy of the" greedy" algorithm. In Proc. 7th Southeastern Conf. on Combinatorics, Graph Theory and Computing, pages 341-350, 1976. Google Scholar
  34. Frank Kammer and Torsten Tholey. Approximation algorithms for intersection graphs. Algorithmica, 68(2):312-336, 2014. Google Scholar
  35. Roie Levin and David Wajc. Streaming submodular matching meets the primal-dual method. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1914-1933. SIAM, 2021. Google Scholar
  36. G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Math. Prog., 14(1):265-294, 1978. Google Scholar
  37. Ami Paz and Gregory Schwartzman. A (2+ ε)-approximation for maximum weight matching in the semi-streaming model. ACM Transactions on Algorithms (TALG), 15(2):1-15, 2018. Google Scholar
  38. Rom Pinchasi. A finite family of pseudodiscs must include a “small” pseudodisc. SIAM Journal on Discrete Mathematics, 28(4):1930-1934, 2014. Google Scholar
  39. Jan Vondrák. Submodularity in combinatorial optimization. PhD thesis, Univerzita Karlova, Matematicko-fyzikální fakulta, 2007. Google Scholar
  40. Yuli Ye and Allan Borodin. Elimination graphs. ACM Transactions on Algorithms (TALG), 8(2):1-23, 2012. Google Scholar
  41. David Zuckerman. 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, pages 681-690, 2006. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact