Abstract
We exhibit an analogy between the problem of pushing forward measurable sets under measure preserving maps and linear relaxations in combinatorial optimization. We show how invariance of hyperfiniteness of graphings under local isomorphism can be reformulated as an infinite version of a natural combinatorial optimization problem, and how one can prove it by extending wellknown proof techniques (linear relaxation, greedy algorithm, linear programming duality) from the finite case to the infinite.
Similar content being viewed by others
References
D. Aldous and R. Lyons, Processes on unimodular random networks, Electron. J. Probab., 12 (2007), Paper 54, 1454–1508
Alon, N., Fischer, E., Krivelevich, M., Szegedy, M.: Efficient testing of large graphs. Combinatorica 20, 451–476 (2000)
Benjamini, I., Schramm, O.: Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126, 565–587 (1996)
Benjamini, I., Schramm, O., Shapira, A.: Every minor-closed property of sparse graphs is testable. Adv. in Math. 223, 2200–2218 (2010)
Elek, G.: On limits of finite graphs. Combinatorica 27, 503–507 (2007)
Elek, G.: Finite graphs and amenability. J. Func. Anal. 263, 2593–2614 (2012)
Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19, 175–220 (1999)
A. Hassidim, J. A. Kelner, H. N. Nguyen and K. Onak, Local graph partitions for approximation and testing, in: Proc. 50\(^{th}\) Ann. IEEE Symp. on Found. Comp. Science (2009), pp. 22–31
Hatami, H., Lovász, L., Szegedy, B.: Limits of locally-globally convergent graph sequences. Geom. Func. Anal. 24, 269–296 (2014)
V. Kaimanovich, Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci. Paris Sr. I Math., 325 (1997), 999–1004
Lovász, L.: On the ratio of optimal fractional and integral covers. Discrete Math. 13, 383–390 (1975)
L. Lovász, Large Networks and Graph Limits, Amer. Math. Soc. (Providence, RI, 2012)
Lovász, L.: Compact graphings. Acta. Math. Hungar. 161, 185–196 (2020)
Schramm, O.: Hyperfinite graph limits, Elect. Res. Announce. Math. Sci. 15, 17–23 (2008)
Scott, A.: Szemerédi's regularity lemma for matrices and sparse graphs. Combin. Prob. Comput. 20, 455–466 (2011)
Szemerédi, E.: On sets of integers containing no \(k\) elements in arithmetic progression. Acta Arithmetica 27, 199–245 (1975)
E. Szemerédi, Regular partitions of graphs, in: Colloque Inter. CNRS J.-C. Bermond, J.-C. Fournier, M. Las Vergnas and D. Sotteau, Eds., (1978), pp. 399–401
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Endre Szemerédi at the occasion of his 80th birthday
Rights and permissions
About this article
Cite this article
Lovász, L. Hyperfinite graphings and combinatorial optimization. Acta Math. Hungar. 161, 516–539 (2020). https://doi.org/10.1007/s10474-020-01065-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01065-y