Low-rank quadratic optimization paper shows million-variable problems can be approximated with constant-size sampling
fpedregosa · x · 2026-07-21
Paper: low-rank structure makes a million-variable discrete quadratic problem tractable
The paper studies maximizing a complex-valued quadratic form over K-th roots of unity. Its main result is that when the quadratic objective matrix has rank r, the global maximizer belongs to a candidate set of size O(r n^{2r-1}), which can be constructed deterministically in O(r n^{2r+1}) time by enumerating vertices of a hyperplane arrangement in R^{2r}.
Key claims
- The algorithm is embarrassingly parallel; with P processors, runtime drops to O(r n^{2r+1} / P).
- For approximately low-rank matrices, a spectral truncation step gives a multiplicative approximation guarantee of 1 - O(||H||₂ / δ), where δ is the eigengap of the underlying rank-r matrix.
- A randomized sampling variant shows that choosing S ≥ O(1 / ε^{r-1}) candidates yields a (1 - ε) cos²(π/K) approximation with high probability.
- Crucially, S is independent of n, reducing total runtime to O(S · n²).
Empirics
- Experiments on synthetic benchmarks and large-scale graphs for MAX-3-CUT reportedly match or exceed SDP solution quality on structured instances.
- The method scales to problems with n ≥ 10⁶ and benefits from massive parallelization on heterogeneous hardware.
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