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Abstract Given a graph of degree over vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth , we develop a local message passing algorithm whose complexity is , and that achieves near optimal cut values among all ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value , where is the value of the Parisi formula from spin glass theory, and (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes after removing a small fraction of vertices. Earlier work established that, for random ‐regular graphs, the typical max‐cut value is . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.
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Low-Degree Hardness of Random Optimization Problems
We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Such problems arise widely in the theory of random graphs, theoretical computer science, and statistical physics. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model, and (b) finding a large independent set in a sparse Erdos-Renyi graph. Two families of algorithms are considered: (a) low-degree polynomials of the input-a general framework that captures methods such as approximate message passing and local algorithms on sparse graphs, among others; and (b) the Langevin dynamics algorithm, a canonical Monte Carlo analogue of the gradient descent algorithm (applicable only for the spherical p-spin glass Hamiltonian). We show that neither family of algorithms can produce nearly optimal solutions with high probability. Our proof uses the fact that both models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objective values are above a certain threshold are either close or far from each other. The crux of our proof is the stability of both algorithms: a small perturbation of the input induces a small perturbation of the output. By an interpolation argument, such a stable algorithm cannot overcome the OGP barrier. The stability of the Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials is established using concepts from Gaussian and Boolean Fourier analysis, including noise sensitivity, hypercontractivity, and total influence.
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- Award ID(s):
- 1712730
- PAR ID:
- 10259980
- Date Published:
- Journal Name:
- 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- Page Range / eLocation ID:
- 131 to 140
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/FOCS46700.2020.00021
