Conjecture on the maximum cut and bisection width in random regular graphs

Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size. Maximum cut and minimal bisection are two famous NP-complete problems with no known general relation between them, hence our conjecture is a surprising property of random regular graphs. We further support the conjecture with numerical simulations. A rigorous proof of this relation is obviously a challenge.

💡 Research Summary

The paper puts forward a striking conjecture about random regular graphs: in the limit of large graph size, the size of a maximum cut (Max‑Cut) equals the total number of edges minus the size of a minimum bisection (Min‑Bisection), up to lower‑order corrections. Formally, |MC| = |E| − |BW| + o(|BW|). This relationship is surprising because Max‑Cut and Min‑Bisection are both classic NP‑complete problems with no known general connection.

To explain the conjecture, the authors recast both optimization problems as ground‑state problems of an Ising spin system defined on the graph. The Hamiltonian is H = −∑{(ij)∈E} J{ij} S_i S_j, where spins S_i∈{±1}. For Max‑Cut the couplings are antiferromagnetic (J_{ij}=−1), for Min‑Bisection they are ferromagnetic (J_{ij}=+1) together with a global magnetization constraint Σ_i S_i = 0. The sizes of the cut and the bisection width can be expressed directly through the ground‑state energy E_GS: |BW| = |E| + E_GS/2, |MC| = |E| − E_GS/2.

The key theoretical claim is that, for a random regular graph, the ground‑state energy does not depend on the fraction ρ of antiferromagnetic edges when the magnetization is forced to zero. The authors introduce a bimodal distribution of couplings, P(J)=ρ δ(J+1)+(1−ρ) δ(J−1), and argue that, because random regular graphs are locally tree‑like, a gauge transformation (σ_i ∈ {±1}) can flip any subset of couplings to −1 without changing the Hamiltonian, provided the boundary conditions have zero total magnetization. In a regular graph all vertices have the same degree, so the only way to maintain zero magnetization after the gauge transformation is for the boundary spins to be balanced, which forces the distribution of cavity fields (the “beliefs” in belief‑propagation language) to be symmetric around zero. Consequently, the cavity equations (or Bethe‑Peierls equations) that determine the ground state are identical for any ρ∈