Document Open Access Logo

Triangles Improve 0.878 Approximation for Maxcut

Authors Fredie George , Anand Louis , Rameesh Paul

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.27.pdf
  • Filesize: 1.05 MB
  • 25 pages
HTML5 Logo HTML (experimental)
LIPIcs.APPROX-RANDOM.2025.27.html

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • Approximation Algorithms
  • Maxcut
  • Semidefinite Programming
  • Edge-disjoint Triangles
  • Unit Ball Graphs
  • Spectral Triadic Graphs

Metrics

Abstract

Maxcut is a fundamental problem in graph algorithms, extensively studied for its theoretical and practical significance. The goal is to partition the vertex set of a graph G = (V, E) into disjoint subsets S and V⧵S so as to maximize the number of edges crossing the cut (S,V⧵S). The seminal work of Goemans and Williamson [Goemans and Williamson, 1995] introduced a semidefinite programming (SDP) based algorithm achieving a α_{GW} ≈ 0.87856-approximation for general graphs, guaranteed to be optimal under the Unique Games Conjecture [Khot, 2002; Khot et al., 2007].
We revisit the Goemans–Williamson SDP and prove that the standard Maxcut SDP achieves a (α_{GW} + Ω(1))-approximation whenever the input graph contains Ω(|E|) edge-disjoint triangles. Our analysis builds on classical rounding techniques studied in [Goemans and Williamson, 1995; Zwick, 1999] and introduces a refined understanding of the SDP solution structure in regimes where the previous guarantees are tight. Our result identifies a simple combinatorial property that may be satisfied by many natural graph classes.
As applications, we show that unit ball graphs and graphs satisfying a spectral transitivity condition (as studied in [Gupta et al., 2016; Basu et al., 2024]) meet our structural criterion, and therefore we get better than α_{GW} approximation guarantees for them. Our algorithm runs in nearly linear time 𝒪̃(|E|), offering a more practical alternative to the PTAS of [Jansen et al., 2005] for unit ball graphs, which has exponential dependence on the approximation parameter.

Cite As Get BibTex

Fredie George, Anand Louis, and Rameesh Paul. Triangles Improve 0.878 Approximation for Maxcut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 27:1-27:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.27

Author Details

Fredie George
  • Indian Institute of Science, Bangalore, India
Anand Louis
  • Indian Institute of Science, Bangalore, India
Rameesh Paul
  • Indian Institute of Science, Bangalore, India

Funding

  • Louis, Anand: Supported by the Walmart Center for Tech Excellence at IISc, and SERB award CRG/2023/002896.
  • Paul, Rameesh: Supported by Prime Minister’s Research Fellowship, India.

Acknowledgements

We thank Rahul Saladi for valuable discussions. We thank anonymous reviewers for their insightful comments.

References

  1. Ranendu Adhikary, Kaustav Bose, Satwik Mukherjee, and Bodhayan Roy. Complexity of maximum cut on interval graphs. In 37th International Symposium on Computational Geometry (SoCG), volume 189, pages 7:1-7:11, 2021. URL: https://doi.org/10.4230/LIPICS.SOCG.2021.7.
  2. Noga Alon and Assaf Naor. Approximating the cut-norm via Grothendieck’s inequality. SIAM J. Comput., 35(4):787-803, 2006. URL: https://doi.org/10.1137/S0097539704441629.
  3. Sanjeev Arora and Satyen Kale. A combinatorial, primal-dual approach to semidefinite programs. J. ACM, 63(2):Art. 12, 35, 2016. URL: https://doi.org/10.1145/2837020.
  4. Sanjeev Arora, David Karger, and Marek Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. J. Comput. System Sci., 58(1):193-210, 1999. URL: https://doi.org/10.1006/jcss.1998.1605.
  5. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2):Art. 5, 37, 2009. URL: https://doi.org/10.1145/1502793.1502794.
  6. Per Austrin, Siavosh Benabbas, and Konstantinos Georgiou. Better balance by being biased: a 0.8776-approximation for Max Bisection. ACM Trans. Algorithms, 13(1):Art. 2, 27, 2016. URL: https://doi.org/10.1145/2907052.
  7. R. Bar-Yehuda, M. M. Halldórsson, J. Naor, H. Shachnai, and I. Shapira. Scheduling split intervals. SIAM J. Comput., 36(1):1-15, 2006. URL: https://doi.org/10.1137/S0097539703437843.
  8. Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999. URL: https://doi.org/10.1126/science.286.5439.509.
  9. Francisco Barahona, Martin Grötschel, Michael Jünger, and Gerhard Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res., 36(3):493-513, June 1988. URL: https://doi.org/10.1287/OPRE.36.3.493.
  10. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science - FOCS 2011, pages 472-481. IEEE Computer Soc., Los Alamitos, CA, 2011. URL: https://doi.org/10.1109/FOCS.2011.95.
  11. Alexey Barsukov and Bodhayan Roy. Maximum cut on interval graphs of interval count two is np-complete, 2024. URL: https://arxiv.org/abs/2203.06630.
  12. Sabyasachi Basu, Suman Kalyan Bera, and C. Seshadhri. Spectral triadic decompositions of real-world networks. SIAM J. Math. Data Sci., 6(3):703-730, 2024. URL: https://doi.org/10.1137/23M1586926.
  13. Piotr Berman and Marek Karpinski. On some tighter inapproximability results (extended abstract). In icalp, pages 200-209, 1999. URL: https://doi.org/10.1007/3-540-48523-6_17.
  14. Hans L. Bodlaender, Celina M. H. de Figueiredo, Marisa Gutierrez, Ton Kloks, and Rolf Niedermeier. Simple max-cut for split-indifference graphs and graphs with few p4’s. In Experimental and Efficient Algorithms, pages 87-99, 2004. URL: https://doi.org/10.1007/978-3-540-24838-5_7.
  15. Hans L. Bodlaender and Klaus Jansen. On the complexity of the maximum cut problem. Nordic Journal of Computing, 7(1):14-31, 2000. Google Scholar
  16. Hans L. Bodlaender, Ton Kloks, and Rolf Niedermeier. Simple max-cut for unit interval graphs and graphs with few p4s. Electronic Notes in Discrete Mathematics, 3:19-26, 1999. URL: https://doi.org/10.1016/S1571-0653(05)80014-9.
  17. Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. System Sci., 13(3):335-379, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80045-1.
  18. Arman Boyacı, Tinaz Ekim, and Mordechai Shalom. A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs. Information Processing Letters, 121:29-33, 2017. URL: https://doi.org/10.1016/j.ipl.2017.01.007.
  19. Arman Boyacı, Tınaz Ekim, and Mordechai Shalom. The maximum cardinality cut problem in co-bipartite chain graphs. Journal of Combinatorial Optimization, 35(1):250-265, 2018. URL: https://doi.org/10.1007/s10878-015-9963-x.
  20. Heinz Breu and David G. Kirkpatrick. Unit disk graph recognition is NP-hard. Comput. Geom., 9(1-2):3-24, 1998. URL: https://doi.org/10.1016/S0925-7721(97)00014-X.
  21. Martin C. Carlisle and Errol L. Lloyd. On the k-coloring of intervals. Discrete Appl. Math., 59(3):225-235, 1995. URL: https://doi.org/10.1016/0166-218X(93)E0174-W.
  22. Charles Carlson, Alexandra Kolla, Ray Li, Nitya Mani, Benny Sudakov, and Luca Trevisan. Lower bounds for max-cut in H-free graphs via semidefinite programming. SIAM J. Discrete Math., 35(3):1557-1568, 2021. URL: https://doi.org/10.1137/20M1333985.
  23. K.C. Chang and D.H.-C. Du. Efficient algorithms for layer assignment problem. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 6(1):67-78, 1987. URL: https://doi.org/10.1109/TCAD.1987.1270247.
  24. Maw-Shang Chang. Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput., 27(6):1671-1694, December 1998. URL: https://doi.org/10.1137/S0097539792238431.
  25. Moses Charikar and Anthony Wirth. Maximizing quadratic programs: Extending grothendieck’s inequality. In focs, pages 54-60, 2004. URL: https://doi.org/10.1109/FOCS.2004.39.
  26. John H. Conway and Neil J. A. Sloane. Sphere Packings, Lattices and Groups, volume 290 of Grundlehren der mathematischen Wissenschaften. Springer, 3rd edition, 1999. Google Scholar
  27. Hervé Daudé, Conrado Martínez, Vonjy Rasendrahasina, and Vlady Ravelomanana. The MAX-CUT of sparse random graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pages 265-271. ACM, New York, 2012. URL: https://doi.org/10.1137/1.9781611973099.24.
  28. Celina MH de Figueiredo, Alexsander A de Melo, Fabiano S Oliveira, and Ana Silva. Maximum cut on interval graphs of interval count four is np-complete. arXiv preprint, 2020. URL: https://arxiv.org/abs/2012.09804.
  29. Charles Delorme and Svatopluk Poljak. Combinatorial properties and the complexity of a max-cut approximation. European J. Combin., 14(4):313-333, 1993. URL: https://doi.org/10.1006/eujc.1993.1035.
  30. Josep Díaz and Marcin Kamiński. Max-cut and max-bisection are NP-hard on unit disk graphs. Theoretical Computer Science, 377(1):271-276, 2007. URL: https://doi.org/10.1016/j.tcs.2007.02.013.
  31. C. S. Edwards. An improved lower bound for the number of edges in a largest bipartite subgraph. In Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pages 167-181. Academia, Prague, 1975. Google Scholar
  32. Pál Erdős. On even subgraphs of graphs. Mat. Lapok, 18:283-288, 1967. Google Scholar
  33. Tomás Feder and Carlos S. Subi. Packing edge-disjoint triangles in given graphs. Electron. Colloquium Comput. Complex., TR12-013, 2012. URL: https://eccc.weizmann.ac.il/report/2012/013.
  34. Uriel Feige, Marek Karpinski, and Michael Langberg. Improved approximation of Max-Cut on graphs of bounded degree. J. Algorithms, 43(2):201-219, 2002. URL: https://doi.org/10.1016/S0196-6774(02)00005-6.
  35. Uriel Feige and Michael Langberg. The RPR² rounding technique for semidefinite programs. J. Algorithms, 60(1):1-23, 2006. URL: https://doi.org/10.1016/j.jalgor.2004.11.003.
  36. Uriel Feige and Gideon Schechtman. On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures Algorithms, 20(3):403-440, 2002. Probabilistic methods in combinatorial optimization. URL: https://doi.org/10.1002/rsa.10036.
  37. Mikael Florén. Approximation of max-cut on graphs of bounded degree, 2016. Google Scholar
  38. David Gamarnik and Quan Li. On the max-cut of sparse random graphs. Random Structures Algorithms, 52(2):219-262, 2018. URL: https://doi.org/10.1002/rsa.20738.
  39. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach., 42(6):1115-1145, 1995. URL: https://doi.org/10.1145/227683.227684.
  40. Rishi Gupta, Tim Roughgarden, and C. Seshadhri. Decompositions of triangle-dense graphs. SIAM J. Comput., 45(2):197-215, 2016. URL: https://doi.org/10.1137/140955331.
  41. Venkatesan Guruswami. Maximum cut on line and total graphs. Discrete Applied Mathematics, 92(2):217-221, 1999. URL: https://doi.org/10.1016/S0166-218X(99)00056-6.
  42. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graphs partitioning and quadratic integer programming with PSD objectives (extended abstract). In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science - FOCS 2011, pages 482-491. IEEE Computer Soc., Los Alamitos, CA, 2011. URL: https://doi.org/10.1109/FOCS.2011.36.
  43. Johan Håstad. Some optimal inapproximability results. In STOC '97 (El Paso, TX), pages 1-10. ACM, New York, 1999. Google Scholar
  44. F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing, 4(3):221-225, 1975. URL: https://doi.org/10.1137/0204019.
  45. Jun-Ting Hsieh and Pravesh K. Kothari. Approximating max-cut on bounded degree graphs: tighter analysis of the FKL algorithm. In 50th International Colloquium on Automata, Languages, and Programming, volume 261 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 77, 7. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.77.
  46. M.L. Huson and A. Sen. Broadcast scheduling algorithms for radio networks. In Proceedings of MILCOM '95, volume 2, pages 647-651 vol.2, 1995. URL: https://doi.org/10.1109/MILCOM.1995.483546.
  47. Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms, 4(4):310-323, 1983. URL: https://doi.org/10.1016/0196-6774(83)90012-3.
  48. Klaus Jansen, Marek Karpinski, Andrzej Lingas, and Eike Seidel. Polynomial time approximation schemes for max-bisection on planar and geometric graphs. SIAM J. Comput., 35(1):110-119, 2005. URL: https://doi.org/10.1137/S009753970139567X.
  49. David S Johnson. The NP-completeness column: an ongoing guide. Journal of Algorithms, 6(3):434-451, 1985. URL: https://doi.org/10.1016/0196-6774(85)90012-4.
  50. John R. Jungck, Gregg Dick, and Amy Gleason Dick. Computer-assisted sequencing, interval graphs, and molecular evolution. Biosystems, 15(3):259-273, 1982. URL: https://doi.org/10.1016/0303-2647(82)90010-7.
  51. G. A. Kabatiansky and V. I. Levenshtein. On bounds for packings on a sphere and in space. Problems of Information Transmission, 14(1):1-17, 1978. Originally in Russian: Probl. Peredachi Inf. 14(1):3-25. Google Scholar
  52. Satyen Kale and C. Seshadhri. Combinatorial approximation algorithms for maxcut using random walks. In Proceedings of 2nd Symposium on Innovations in Computer Science (ICS), pages 367-388, 2011. URL: http://conference.iiis.tsinghua.edu.cn/ICS2011/content/papers/20.html.
  53. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), The IBM Research Symposia Series, pages 85-103. Plenum, New York-London, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  54. Ken-ichi Kawarabayashi, Mikkel Thorup, and Hirotaka Yoneda. Better coloring of 3-colorable graphs. In STOC'24 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 331-339. ACM, New York, 2024. URL: https://doi.org/10.1145/3618260.3649768.
  55. J. Mark Keil. Finding Hamiltonian circuits in interval graphs. Inform. Process. Lett., 20(4):201-206, 1985. URL: https://doi.org/10.1016/0020-0190(85)90050-X.
  56. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pages 767-775. ACM, New York, 2002. URL: https://doi.org/10.1145/509907.510017.
  57. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput., 37(1):319-357, 2007. URL: https://doi.org/10.1137/S0097539705447372.
  58. Jan Kratochvíl, Tomáš Masařík, and Jana Novotná. U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020), volume 170, pages 57:1-57:14, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.57.
  59. Madhav V. Marathe, R. Ravi, and C. Pandu Rangan. Generalized vertex covering in interval graphs. Discrete Appl. Math., 39(1):87-93, 1992. URL: https://doi.org/10.1016/0166-218X(92)90116-R.
  60. Claire Mathieu and Warren Schudy. Yet another algorithm for dense max cut: go greedy. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 176-182. ACM, New York, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347102.
  61. Renee Mirka and David P. Williamson. An experimental evaluation of semidefinite programming and spectral algorithms for max cut. ACM J. Exp. Algorithmics, 28, August 2023. URL: https://doi.org/10.1145/3609426.
  62. Andrea Montanari and Subhabrata Sen. Semidefinite programs on sparse random graphs and their application to community detection. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, pages 814-827, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2897518.2897548.
  63. Yurii Nesterov. Semidefinite relaxation and nonconvex quadratic optimization. Optimization Methods & Software, 9:141-160, 1998. URL: https://api.semanticscholar.org/CorpusID:121309892.
  64. Ryan O'Donnell and Yi Wu. An optimal SDP algorithm for max-cut, and equally optimal long code tests. In STOC'08, pages 335-344. ACM, New York, 2008. URL: https://doi.org/10.1145/1374376.1374425.
  65. Prasad Raghavendra and David Steurer. How to round any CSP. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2009, pages 586-594. IEEE Computer Soc., Los Alamitos, CA, 2009. URL: https://doi.org/10.1109/FOCS.2009.74.
  66. Sartaj Sahni and Teofilo Gonzalez. P-complete approximation problems. J. Assoc. Comput. Mach., 23(3):555-565, 1976. URL: https://doi.org/10.1145/321958.321975.
  67. José A. Soto. Improved analysis of a max-cut algorithm based on spectral partitioning. SIAM J. Discrete Math., 29(1):259-268, 2015. URL: https://doi.org/10.1137/14099098X.
  68. David Steurer. Fast SDP algorithms for constraint satisfaction problems. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 684-697. SIAM, Philadelphia, PA, 2010. URL: https://doi.org/10.1137/1.9781611973075.56.
  69. Chaitanya Swamy. Correlation clustering: maximizing agreements via semidefinite programming. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 526-527. ACM, New York, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982866.
  70. Luca Trevisan. Max cut and the smallest eigenvalue. siamj, 41(6):1769-1786, 2012. URL: https://doi.org/10.1137/090773714.
  71. Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, and David P. Williamson. Gadgets, approximation, and linear programming. SIAM J. Comput., 29(6):2074-2097, 2000. URL: https://doi.org/10.1137/S0097539797328847.
  72. Martin J. Wainwright. High-dimensional statistics, volume 48 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2019. A non-asymptotic viewpoint. URL: https://doi.org/10.1017/9781108627771.
  73. Stanley Wasserman and Katherine Faust. Social Network Analysis: Methods and Applications. Structural Analysis in the Social Sciences. Cambridge University Press, 1994. URL: https://doi.org/10.1017/CBO9780511815478.
  74. Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393(6684):440-442, 1998. URL: https://doi.org/10.1038/30918.
  75. Mihalis Yannakakis and Fănică Gavril. The maximum k-colorable subgraph problem for chordal graphs. Inform. Process. Lett., 24(2):133-137, 1987. URL: https://doi.org/10.1016/0020-0190(87)90107-4.
  76. Uri Zwick. Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), pages 679-687. ACM, New York, 1999. URL: https://doi.org/10.1145/301250.301431.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact