## The Annals of Probability

### Extremal cuts of sparse random graphs

#### Abstract

For Erdős–Rényi random graphs with average degree $\gamma$, and uniformly random $\gamma$-regular graph on $n$ vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are $n(\frac{\gamma}{4}+\mathsf{P}_{*}\sqrt{\frac{\gamma}{4}}+o(\sqrt{\gamma}))+o(n)$ while the size of the minimum bisection is $n(\frac{\gamma}{4}-\mathsf{P}_{*}\sqrt{\frac{\gamma}{4}}+o(\sqrt{\gamma}))+o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington–Kirkpatrick model, with $\mathsf{P}_{*}\approx0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi’s formula.

#### Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 1190-1217.

Dates
Revised: August 2015
First available in Project Euclid: 31 March 2017

https://projecteuclid.org/euclid.aop/1490947317

Digital Object Identifier
doi:10.1214/15-AOP1084

Mathematical Reviews number (MathSciNet)
MR3630296

Zentralblatt MATH identifier
1372.05196

#### Citation

Dembo, Amir; Montanari, Andrea; Sen, Subhabrata. Extremal cuts of sparse random graphs. Ann. Probab. 45 (2017), no. 2, 1190--1217. doi:10.1214/15-AOP1084. https://projecteuclid.org/euclid.aop/1490947317

#### References

• [1] Alon, N. (1997). On the edge-expansion of graphs. Combin. Probab. Comput. 6 145–152.
• [2] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2009). An Introduction to Random Matrices. Cambridge Univ. Press, Cambridge.
• [3] Auffinger, A. and Chen, W. (2015). The Parisi formula has a unique minimizer. Comm. Math. Phys. 335 1429–1444.
• [4] Bayati, M., Gamarnik, D. and Tetali, P. (2013). Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Ann. Probab. 41 4080–4115.
• [5] Bollobás, B. (1988). The isoperimetric number of random regular graphs. European J. Combin. 9 241–244.
• [6] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
• [7] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
• [8] Chvátal, V. (1979). The tail of the hypergeometric distribution. Discrete Math. 25 285–287.
• [9] Coja-Oghlan, A. (2007). On the Laplacian eigenvalues of $G_{n,p}$. Combin. Probab. Comput. 16 923–946.
• [10] Coppersmith, D., Gamarnik, D., Hajiaghayi, M. T. and Sorkin, G. B. (2004). Random MAXSAT, random MAXCUT, and their phase transitions. Random Structures Algorithms 24 502–545.
• [11] Crisanti, A. and Rizzo, T. (2002). Analysis of the $\infty$-replica symmetry breaking solution of the Sherrington–Kirkpatrick model. Phys. Rev. E (3) 65 046137, 9.
• [12] Daudé, H., Martínez, C., Rasendrahasina, V. and Ravelomanana, V. (2012). The MAX-CUT of sparse random graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms 265–271. ACM, New York.
• [13] Decelle, A., Krzakala, F., Moore, C. and Zdeborová, L. (2011). Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E (3) 84 066106.
• [14] Díaz, J., Petit, J. and Serna, M. J. (2002). A survey on graph layout problems. ACM Comput. Surv. 34 313–356.
• [15] Díaz, J., Serna, M. J. and Wormald, N. C. (2004). Computation of the bisection width for random $d$-regular graphs. In LATIN 2004: Theoretical Informatics. Lecture Notes in Computer Science 2976 49–58. Springer, Berlin.
• [16] Feige, U. and Krauthgamer, R. (2000). A polylogarithmic approximation of the minimum bisection. In Foundations of Computer Science 105–115. Redondo Beach, CA.
• [17] Feige, U. and Ofek, E. (2005). Spectral techniques applied to sparse random graphs. Random Structures Algorithms 27 251–275.
• [18] Franz, S. and Leone, M. (2003). Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111 535–564.
• [19] Friedman, J. (2003). A proof of Alon’s second eigenvalue conjecture. In Proceedings of the 35th Symposium on Theory of Computing 720–724. San Diego, CA.
• [20] Frieze, A. M. and Łuczak, T. (1992). On the independence and chromatic numbers of random regular graphs. J. Combin. Theory Ser. B 54 123–132.
• [21] Gamarnik, D. and Li, Q. (2014). On the Max-Cut over sparse random graph. Available at arXiv:1411.1698v1.
• [22] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
• [23] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
• [24] Guerra, F. and Toninelli, F. L. (2004). The high temperature region of the Viana–Bray diluted spin glass model. J. Stat. Phys. 115 531–555.
• [25] Halperin, E. and Zwick, U. (2002). A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems. Random Structures Algorithms 20 382–402.
• [26] Hastad, J. (1999). Some optimal inapproximability results. In STOC ’97 (el Paso, TX) 1–10. ACM, New York.
• [27] Hofstad, R. (2009). Random graphs and complex networks. Preprint. Available at http://www.win.tue.nl/~rhofstad/.
• [28] Jagannath, A. and Tobasco, I. (2016). Dynamic programming approach to the Parisi variational problem. Proc. Amer. Math. Soc. 144 3135–3150.
• [29] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
• [30] Khorunzhiy, O., Kirsch, W. and Müller, P. (2006). Lifshitz tails for spectra of Erdős–Rényi random graphs. Ann. Appl. Probab. 16 295–309.
• [31] Khot, S. (2004). Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In Foundations of Computer Science 136–145. Roma, Italy.
• [32] Kim, J. H. (2006). Poisson cloning model for random graphs. In International Congress of Mathematicians. Vol. III 873–897. Eur. Math. Soc., Zürich.
• [33] Luczak, M. J. and McDiarmid, C. (2001). Bisecting sparse random graphs. Random Structures Algorithms 18 31–38.
• [34] Massoulié, L. (2014). Community detection thresholds and the weak Ramanujan property. In STOC’14—Proceedings of the 2014 ACM Symposium on Theory of Computing 694–703. ACM, New York.
• [35] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
• [36] Mézard, M., Parisi, G. and Virasoro, M. (1986). Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. World Scientific Lecture Notes in Physics 9. World Scientific, Singapore.
• [37] Montanari, A. and Sen, S. (2015). Semidefinite programs on sparse random graphs and their application to community detection. Available at arXiv:1504.05910.
• [38] Mossel, E., Neeman, J. and Sly, A. (2013). A proof of the block model threshold conjecture. Available at arXiv:1311.4115.
• [39] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer, New York.
• [40] Percus, A. G., Istrate, G., Gonçalves, B., Sumi, R. Z. and Boettcher, S. (2008). The peculiar phase structure of random graph bisection. J. Math. Phys. 49 125219, 13.
• [41] Poljak, S. and Tuza, Z. (1995). Maximum cuts and large bipartite subgraphs. In Combinatorial Optimization (New Brunswick, NJ, 19921993). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 20 181–244. Amer. Math. Soc., Providence, RI.
• [42] Schmidt, M. J. (2008). Replica symmetry breaking at low temperatures. Ph.D. thesis, Univ. Würzburg.
• [43] Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 46. Springer, Berlin.
• [44] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
• [45] Wormald, N. C. (1999). Models of random regular graphs. In Surveys in Combinatorics, 1999 (Canterbury). London Mathematical Society Lecture Note Series 267 239–298. Cambridge Univ. Press, Cambridge.
• [46] Zdeborová, L. and Boettcher, S. (2010). A conjecture on the maximum cut and bisection width in random regular graphs. J. Stat. Mech. Theory Exp. 2010 P02020.