Develop an optimal message-passing algorithm for Max-$k$-Cut

Construct a local message-passing scheme for Max-$k$-Cut on locally tree-like regular graphs and prove whether it attains the Potts–Parisi optimum $P_k^{\star}$ at leading order, thereby determining whether such a scheme can match or surpass the guarantees of the Local Vector algorithm.

Background

The paper notes that message-passing algorithms attain optimal or near-optimal leading-order performance for related binary optimization problems, including the Sherrington–Kirkpatrick model and Max-Cut on locally tree-like regular graphs. It leaves unresolved whether an analogous algorithm exists for the kk-label setting and whether the resulting performance reaches the mean-field Potts antiferromagnet benchmark. A positive result would yield a classically optimal leading-order algorithm for Max-kk-Cut.

References

The second open question is algorithmic. For the Sherrington--Kirkpatrick model, Montanari gave a message-passing algorithm that reaches the Parisi optimum whenever no overlap gap is present, and El Alaoui, Montanari, and Sellke constructed a local message-passing algorithm that achieves the near-optimal cut value on locally tree-like regular graphs for Max-Cut. It is natural to ask whether a message-passing scheme of this kind can be designed for Max-$k$-Cut and whether it provably attains the Potts--Parisi optimum $P{\star}_k$, either matching or surpassing the guarantees established here.

— Provable Classical and Quantum Local Algorithms for Max-$k$-Cut and Quantum Advantage at Moderate Girth  (2609.39042 - Apte et al., 30 Sep 2026) in Section 6, “Conclusion and Future Work”