Deterministic approximation factors for the nonnegative permanent
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For a nonnegative rational n×n matrix, can a deterministic algorithm approximate its permanent within cⁿ for an absolute c<√2 in polynomial time in the binary input length?
Why it matters and what counts as progress
Why it matters: A smaller exponential base improves the general nonnegative-matrix guarantee beyond the Bethe-based (√2)ⁿ benchmark.
What counts as progress: A two-sided approximation with a proved base below √2 and binary polynomial runtime, or a stronger lower bound for this approximation task.
How the problem developed
- 2011 · Lower bound [1]
Leonid Gurvits Established the Bethe approximation as a lower bound for the permanent of a nonnegative matrix.
- 2018 · Upper bound [2]
Nima Anari and Alireza Rezaei Proved the sharp universal comparison per(A) ≤ 2^{n/2} Bethe(A), yielding a deterministic approximation baseline but not the improved exponent sought here.
- 1 Sept 2026 · Reported claim [3]
Nima Anari, with ChatGPT 5.6 Sol Pro The preprint gives a deterministic polynomial-time c^n approximation for every nonnegative matrix, for a constant c<√2.
AI contribution
AI-developed proof improves the permanent-approximation base
For every nonnegative rational n×n matrix, Anari’s formal development constructs one rational-output algorithm and an absolute constant c<√2 with a two-sided cⁿ approximation guarantee for the permanent. It also proves deterministic polynomial time in the ordinary binary input size.
Sources
- Leonid Gurvits, Unleashing the power of Schrijver’s permanental inequality with the help of the Bethe Approximation
- Nima Anari and Alireza Rezaei, A Tight Analysis of Bethe Approximation for Permanent
- Nima Anari, Beyond the Bethe Approximation of the Permanent
- Beyond Bethe: theorem, disclosure and review
- Beyond Bethe: formal theorem statement
Website timeline
Publication dates and editorial changes.
AI-developed proof improves the permanent-approximation base