Graph Coarsening Enables Escaping Local Optima in Compute-Constrained Nonconvex MAX-CUT

A combined CMG+PPR pipeline reaches 97.5% of the known exact optimum on average and 98.4% on its best run on the t2g20 benchmark instance (n=400), according to arXiv:2610.19043v1. The method warm-starts Particle Polynomial Relaxation (PPR), a moment-based nonconvex continuous reformulation, with Combinatorial Multigrid (CMG), a spectral graph-coarsening technique that produces structure-aware initial solutions in near-linear time without training data. An extended variant with discrete local-search refinement reaches the published optimum on all sixty Biq Mac instances evaluated by a recent neural branch-and-bound method, and a sparse CSR representation scales the approach to G-set instances with up to 20,000 vertices while cutting memory requirements by up to 3333x.

arXiv:2610.19043v1 Announce Type: new Abstract: Because its binary optimization landscape contains exponentially many local optima, MAX-CUT is highly sensitive to initialization. We study Particle Polynomial Relaxation PPR , a moment-based nonconvex continuous reformulation possessing an advantageous optimization landscape. We warm-start PPR with Combinatorial Multigrid CMG , a spectral graph-coarsening method that produces structure-aware initial solutions in near-linear time without training data. On \texttt{t2g20} $n=400$ , benchmark instance with known exact optimum, the combined CMG+PPR pipeline reaches $97.5\%$ of optimum on average and $98.4\%$ on the best run. An extended variant with discrete local-search refinement reaches the published optimum on all sixty Biq Mac instances evaluated by a recent neural branch-and-bound method. Using a sparse CSR representation, the same approach scales to G-set instances with up to $20{,}000$ vertices reducing memory requirements by up to $3333\times$ while maintaining a consistent improvement over the CMG warm start. Gains are largest on sparse weighted graphs; on dense graphs we complement the pipeline with spectral sparsification.