Abstract
For a graph G, let f(G) denote the size of the maximum cut in G. The problem of estimating f(G) as a function of the number of vertices and edges of G has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming (SDP), to prove lower bounds on f(G). We use this approach to find large cuts in graphs with few triangles and in \(K_r\)-free graphs.
Alexandra Kolla was supported by NSF CAREER grant 1452923 as well as NSF AF grant 1814385.
Ray Li was supported by an NSF GRF grant DGE-1656518 and by NSF grant CCF-1814629.
Nitya Mani was supported in part by a Stanford Undergraduate Advising and Research Major Grant.
Luca Trevisan was supported by the NSF under grant CCF 181543 and his work on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 834861).
Benny Sudakov was supported in part by SNSF grant 200021_196965.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Alon, N.: Bipartite subgraphs. Combinatorica 16, 301–311 (1996)
Alon, N., Bollobás, B., Krivelevich, M., Sudakov, B.: Maximum cuts and judicious partitions in graphs without short cycles. J. Combin. Theory Ser. B 88, 329–346 (2003)
Alon, N., Krivelevich, M., Sudakov, B.: Max cut in \(H\)-free graphs. Combin. Prob. Comput. 14, 629–647 (2005)
Alon, N., Halperin, E.: Bipartite subgraphs of integer weighted graphs. Discret. Math. 181, 19–29 (1998)
Bollobás, B., Scott, A.D.: Better bounds for max cut. In: Bollobás, B. (ed.) Contemporary Combinatorics. Bolyai Society Mathematical Studies, pp. 185–246. Springer, Heidelberg (2002)
Conlon, D., Fox, J., Kwan, M., Sudakov, B.: Hypergraph cuts above the average. Isr. J. Math. 233(1), 67–111 (2019). https://doi.org/10.1007/s11856-019-1897-z
Edwards, C.S.: Some extremal properties of bipartite subgraphs. Canad. J. Math. 3, 475–485 (1973)
Edwards, C.S.: An improved lower bound for the number of edges in a largest bipartite subgraph. In: Proceedings of the 2nd Czechoslovak Symposium on Graph Theory, Prague, pp. 167–181 (1975)
Erdős, P.: On even subgraphs of graphs. Mat. Lapok 18, 283–288 (1967)
Erdős, P.: Problems and results in graph theory and combinatorial analysis. In: Proceedings of the 5th British Combinatorial Conference, pp. 169–192 (1975)
Erdős, P., Faudree, R., Pach, J., Spencer, J.: How to make a graph bipartite. J. Combin. Theory Ser. B 45, 86–98 (1988)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)
O’Donnell, R., Wu, Y.: An optimal SDP algorithm for Max-Cut, and equally optimal long code tests. In: Proceedings of the 40th ACM Symposium on Theory of Computing, pp. 335–344 (2008)
Poljak, S. Tuza, Zs.: Bipartite subgraphs of triangle-free graphs. SIAM J. Discret. Math. 7, 307–313 (1994)
Shearer, J.: A note on bipartite subgraphs of triangle-free graphs. Rand. Struct. Alg. 3, 223–226 (1992)
Sudakov, B.: Making a \(K_4\)-free graph bipartite. Combinatorica 27, 509–518 (2007)
Acknowledgements
The authors thank Jacob Fox and Matthew Kwan for helpful discussions and feedback. The authors thank Joshua Brakensiek for finding an error in an earlier draft of this paper. The authors thank Joshua Brakensiek and Yuval Wigderson for helpful feedback on an earlier draft of the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Carlson, C., Kolla, A., Li, R., Mani, N., Sudakov, B., Trevisan, L. (2020). Lower Bounds for Max-Cut via Semidefinite Programming. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_38
Download citation
DOI: https://doi.org/10.1007/978-3-030-61792-9_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-61791-2
Online ISBN: 978-3-030-61792-9
eBook Packages: Computer ScienceComputer Science (R0)