[Paper Review] On the unbalanced cut problem and the generalized Sherrington-Kirkpatrick model

Aukosh Jagannath, Subhabrata Sen|arXiv (Cornell University)|Jul 27, 2017

Topological and Geometric Data Analysis51 references3 citations

TL;DR

This paper establishes a strict asymptotic inequality showing that the unbalanced cut problem on sparse Erdős-Rényi random graphs yields a strictly larger maximum cut than on random regular graphs of the same average degree, with the difference scaling as $ C(\alpha)\sqrt{d}N $ for $ \alpha < 1/2 $. The result is derived via a variational representation of the ground state energy in generalized $ p $-spin glass models, linking combinatorial optimization to statistical physics through gamma convergence and interpolation techniques.

ABSTRACT

We establish a strict asymptotic inequality between a class of graph partition problems on the sparse End\H{o]s-Rényi and random regular graph ensembles with the same average degree. Along the way, we establish a variational representation for the ground state energy for generalized mixed $p$-spin glasses and derive strict comparison inequalities for such models as the alphabet changes.

Motivation & Objective

• To compare the asymptotic behavior of the unbalanced cut problem across two canonical random graph ensembles: sparse Erdős-Rényi and random regular graphs.
• To establish a strict positive lower bound on the difference in maximum $ \alpha $-cuts between these ensembles in the large-degree limit.
• To formalize a connection between combinatorial graph partitioning and the ground state energy of generalized Sherrington-Kirkpatrick spin glass models.
• To derive variational representations and comparison inequalities for generalized $ p $-spin glass Hamiltonians as the system alphabet changes.
• To provide a rigorous foundation for the mean-field approximation of graph cut problems using tools from statistical mechanics and stochastic analysis.

Proposed method

• Utilizes an interpolating Hamiltonian framework to compare the ground state energies of $ p $-spin glass models on different graph ensembles.
• Applies Gaussian integration by parts to derive a differential inequality for the free energy, showing monotonicity in the interpolation parameter $ t \in [0,1] $.
• Employs a super-additivity argument on the normalized maximum energy over magnetization intervals to prove convergence of the limiting free energy.
• Establishes continuity and concavity of the limiting free energy function via covering arguments and moment bounds on Gaussian processes.
• Uses gamma convergence and variational representations to relate the graph cut problem to the ground state energy of generalized $ p $-spin glasses.
• Applies a limiting procedure as $ \eta \to 0 $ to define the limiting free energy $ G(a) $, which is continuous and concave on $ [-1,1] $.

Experimental results

Research questions

• RQ1 Does the maximum $ \alpha $-cut on sparse Erdős-Rényi graphs exceed that on random regular graphs for $ \alpha < 1/2 $, and by how much?
• RQ2 Can the unbalanced cut problem be rigorously linked to the ground state energy of generalized $ p $-spin glass models?
• RQ3 What is the asymptotic behavior of the difference in maximum cut values between Erdős-Rényi and random regular graphs as the average degree $ d \to \infty $?
• RQ4 Is there a strict positive lower bound on the normalized difference in maximum cut values between the two ensembles?
• RQ5 Can variational representations and comparison inequalities be derived for generalized $ p $-spin glass models under changes in the system's alphabet or configuration space?

Key findings

• For any $ 0 < \alpha < 1/2 $, there exists a constant $ C(\alpha) > 0 $ such that the difference in maximum $ \alpha $-cuts between Erdős-Rényi and random regular graphs satisfies a strict lower bound: \[ \varliminf_{d\to\infty}\varliminf_{N\to\infty}\frac{{\sf MCUT}_{\alpha}(G(N,\frac{d}{N})) - {\sf MCUT}_{\alpha}(G^{\rm Reg}(N,d))}{\sqrt{d}N} \geq C(\alpha) \] almost surely.
• The difference in cut values is strictly positive in the large-degree limit, indicating that the rigidity of random regular graphs leads to suboptimal performance in unbalanced cut problems compared to Erdős-Rényi graphs.
• The ground state energy of the generalized Sherrington-Kirkpatrick model admits a variational representation that allows for strict comparison inequalities as the system's alphabet or configuration space changes.
• The limiting free energy function $ G(a) $, representing the normalized maximum energy over magnetization intervals, is continuous and concave on $ [-1,1] $, with a well-defined pointwise limit as $ \eta \to 0 $.
• The method yields a natural conjecture for the sharp constant $ C(\alpha) $, which is expected to be related to the minimizer of a certain functional $ \mathcal{P}_{T} $, though the paper does not claim this is the exact value.
• The result reverses for the minimum $ \alpha $-cut problem, with the inequality direction flipped, due to the same underlying variational structure.