Graph Coarsening Enables Escaping Local Optima in Compute-Constrained Nonconvex MAX-CUT
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Abstract (arXiv)
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 textttt2g20 (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× 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.
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References · 9
- 2013Ising formulations of many NP problemscited by 207
- 2017Cooperative Game Theory Approaches for Network Partitioningcited by 135
- 2015Parallelizing the dual revised simplex methodcited by 89
- 2016The non-convex Burer-Monteiro approach works on smooth semidefinite programscited by 46
- 2023Are Graph Neural Networks Optimal Approximation Algorithms?cited by 11
- 2017The Mixing method: low-rank coordinate descent for semidefinite programming with diagonal constraintscited by 9
- 2024Learning-Augmented Streaming Algorithms for Approximating MAX-CUTcited by 8
- 2023An Efficient Framework for Global Non-Convex Polynomial Optimization over the Hypercubecited by 1
- 2026Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networkscited by 1
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