Read the equation
The saved representative instance has normalized minimum objective approximately 0.2578092851. This number is the ratio J=Φ_SI/K at the minimizing partition. It is dimensionless and belongs to that particular generated network, rather than a universal information threshold.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| J_star | Minimum normalized objective |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
\subsection*{Execution and an inspectable example}
The 60 instance computations totaled 87.24 seconds, excluding initial validation/JIT warm-up. Execution was serial with one BLAS thread on a Linux x86-64 environment reporting an AMD EPYC 9V74 CPU; Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0 and Numba 0.65.1. Timings are environment-specific.
For the first fixed instance, \texttt{n08\_dense\_r00}, the exact partition is $\{1,2,3,7\}\mid\{4,5,6,8\}$: $J^*=0.2578092851$, $\Phi_{\rm SI}=2.1867917397$ nats and $K=8.4822070674$ nats. Queyranne instead returns $\{1,2,3,6\}\mid\{4,5,7,8\}$ with $J=0.2788420313$. Every other instance, including unsuccessful searches, is retained in the companion data.
\newpage
\section*{6. Interpretation and limits}
\textbf{Normalization materially changes this benchmark.} A separately labeled \emph{post-run diagnostic} found that Queyranne recovered all 60 \emph{unnormalized} stochastic-interaction minima, versus 11 normalized minima. Normalization changed the exact minimizing partition in 37/60 networks. An exhaustive post-run check of 6,804,480 elementary submodularity inequalities found violations in all 60 normalized objective tables. Thus the direct normalized search cannot inherit an exactness guarantee merely from success on a related objective. These post-run diagnostics leave the primary results unchanged.
[Context excerpt; full source linked.]Exact original notation
J^*=0.2578092851