The Annals of Probability

Suboptimality of local algorithms for a class of max-cut problems

Abstract

We show that in random $K$-uniform hypergraphs of constant average degree, for even $K\geq 4$, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain nontrivial interval—a phenomenon referred to as the overlap gap property—which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models and showing the overlap gap property in the latter setting.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1587-1618.

Dates
Revised: March 2018
First available in Project Euclid: 2 May 2019

https://projecteuclid.org/euclid.aop/1556784027

Digital Object Identifier
doi:10.1214/18-AOP1291

Mathematical Reviews number (MathSciNet)
MR3945754

Zentralblatt MATH identifier
07067277

Citation

Chen, Wei-Kuo; Gamarnik, David; Panchenko, Dmitry; Rahman, Mustazee. Suboptimality of local algorithms for a class of max-cut problems. Ann. Probab. 47 (2019), no. 3, 1587--1618. doi:10.1214/18-AOP1291. https://projecteuclid.org/euclid.aop/1556784027

References

• [1] Auffinger, A. and Chen, W.-K. (2015). The Parisi formula has a unique minimizer. Comm. Math. Phys. 335 1429–1444.
• [2] Auffinger, A. and Chen, W.-K. (2017). Parisi formula for the ground state energy in the mixed $p$-spin model. Ann. Probab. 45 4617–4631.
• [3] Auffinger, A., Chen, W.-K. and Zeng, Q. (2017). The SK model is Full-step Replica Symmetry Breaking at zero temperature. Preprint. Available at arXiv:1703.06872.
• [4] Backhausz, A. and Szegedy, B. (2016). On the almost eigenvectors of random regular graphs. Preprint. Available at arXiv:1607.04785.
• [5] Backhausz, A. and Szegedy, B. (2018). On large girth regular graphs and random processes on trees. Random Structures Algorithms. To appear. Available at arXiv:1406.4420.
• [6] Backhausz, Á. and Virág, B. (2017). Spectral measures of factor of i.i.d. processes on vertex-transitive graphs. Ann. Inst. Henri Poincaré Probab. Stat. 53 2260–2278.
• [7] 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.
• [8] Ben Arous, G. and Jagannath, A. (2018). Spectral gap estimates in mean field spin glasses. Comm. Math. Phys. 361 1–52.
• [9] Bruckner, A. (1994). Differentiation of Real Functions, 2nd ed. CRM Monograph Series 5. Amer. Math. Soc., Providence, RI.
• [10] Chen, W.-K. (2017). Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound. Ann. Probab. 45 3929–3966.
• [11] Chen, W.-K., Handschy, M. and Lerman, G. (2018). On the energy landscape of the mixed even $p$-spin model. Probab. Theory Related Fields 171 53–95.
• [12] Chen, W.-K. and Panchenko, D. (2018). Disorder chaos in some diluted spin glass models. Ann. Appl. Probab. 28 1356–1378.
• [13] De Sanctis, L. (2004). Random multi-overlap structures and cavity fields in diluted spin glasses. J. Stat. Phys. 117 785–799.
• [14] Dembo, A., Montanari, A. and Sen, S. (2017). Extremal cuts of sparse random graphs. Ann. Probab. 45 1190–1217.
• [15] Elek, G. and Lippner, G. (2010). Borel oracles. An analytical approach to constant-time algorithms. Proc. Amer. Math. Soc. 138 2939–2947.
• [16] Fan, Z. and Montanari, A. (2017). How well do local algorithms solve semidefinite programs? In STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing 604–614. ACM, New York.
• [17] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
• [18] Franz, S. and Leone, M. (2003). Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111 535–564.
• [19] Gamarnik, D. and Sudan, M. (2017). Limits of local algorithms over sparse random graphs. Ann. Probab. 45 2353–2376.
• [20] Gamarnik, D. and Sudan, M. (2017). Performance of sequential local algorithms for the random NAE-$K$-SAT problem. SIAM J. Comput. 46 590–619.
• [21] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
• [22] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
• [23] 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.
• [24] Harangi, V. and Virág, B. (2015). Independence ratio and random eigenvectors in transitive graphs. Ann. Probab. 43 2810–2840.
• [25] Hatami, H., Lovász, L. and Szegedy, B. (2014). Limits of locally-globally convergent graph sequences. Geom. Funct. Anal. 24 269–296.
• [26] Hoppen, C. and Wormald, N. (2013). Local algorithms, regular graphs of large girth, and random regular graphs. Preprint. Available at arXiv:1308.0266.
• [27] Jagannath, A., Ko, J. and Sen, S. (2018). Max $\kappa$-cut and the inhomogeneous Potts spin glass. Ann. Appl. Probab. 28 1536–1572.
• [28] Jagannath, A. and Tobasco, I. (2016). A dynamic programming approach to the Parisi functional. Proc. Amer. Math. Soc. 144 3135–3150.
• [29] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
• [30] Lyons, R. (2017). Factors of IID on trees. Combin. Probab. Comput. 26 285–300.
• [31] Lyons, R. and Nazarov, F. (2011). Perfect matchings as IID factors on non-amenable groups. European J. Combin. 32 1115–1125.
• [32] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer, New York.
• [33] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
• [34] Panchenko, D. (2014). The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 946–958.
• [35] Panchenko, D. (2018). On the $K$-sat model with large number of clauses. Random Structures Algorithms 52 536–542.
• [36] Panchenko, D. and Talagrand, M. (2004). Bounds for diluted mean-fields spin glass models. Probab. Theory Related Fields 130 319–336.
• [37] Parisi, G. (1980). A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13 L115–L121.
• [38] Parisi, G. (1979). Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43 1754–1756.
• [39] Rahman, M. (2016). Factor of IID percolation on trees. SIAM J. Discrete Math. 30 2217–2242.
• [40] Rahman, M. and Virág, B. (2017). Local algorithms for independent sets are half-optimal. Ann. Probab. 45 1543–1577.
• [41] Sen, S. (2016). Optimization on sparse random hypergraphs and spin glasses. Random Structures Algorithms. To appear. Available at arXiv:1606.02365.
• [42] Talagrand, M. (2011). Mean-Field Models for Spin Glasses. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics 54. Springer, Berlin.
• [43] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
• [44] Talagrand, M. (2007). Mean field models for spin glasses: Some obnoxious problems. In Spin Glasses. Lecture Notes in Math. 1900 63–80. Springer, Berlin.