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NP-completeness of the Minimum Circuit Size Problem (MCSP)

Determine whether the Minimum Circuit Size Problem for Boolean circuits is NP-complete, or prove that it is not NP-complete, thereby settling the complexity status of MCSP.

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Background

The paper reviews classical results on the Minimum Circuit Size Problem (MCSP), noting long-standing interest and partial hardness results. While MCSP is known to be in NP and has various NP-hardness results for related formulations, its precise status as NP-complete remains unresolved. Clarifying this would have broad implications for circuit complexity and the feasibility of algorithmic approaches to circuit minimization.

References

Minimum Circuit Size Problem (MCSP): The problem of minimizing circuit size has been known to be NP for a long time, but it remains unknown whether it is NP complete or not \citep{hitchcock_2015_NP_complete_MCSP,Ilango_2020_multidim_MCSP_NP_hard}.

— Deep Learning as a Convex Paradigm of Computation: Minimizing Circuit Size with ResNets (2511.20888 - Jacot, 25 Nov 2025) in Related Works, Introduction ("Minimum Circuit Size Problem (MCSP)" paragraph)