← 研踪 PaperTrailarXiv:2611.04457 · v1
Entropy-Weighted Soundness for Multi-Answer Birthday Graphs
首次提交 2026-11-03 15:47
版本更新 2026-11-03 15:47
雷达观测 2026-11-04 23:40
摘要
The vertices of a Boolean birthday graph are partial assignments on $\ell$-subsets of $n$ variables. Although a satisfying assignment induces a $K$-clique for $K=\binom n\ell$, an arbitrary selection of $K$ vertices may repeat some views and omit others. This makes soundness substantially harder than under uniform view sampling. For bounded-occurrence, constant-gap Boolean ternary CSPs whose constraints each involve three distinct variables (allowing arbitrary Boolean predicates), we prove that the graph's maximum induced $K$-vertex density satisfies \[ ρ^*(K)\le 2\exp\!\left(-c\, \frac{\ell^3}{n^2\log(en/\ell)}\right) \] for $2\le\ell\le x_0n$ and $\ell\log(en/\ell)\ge C_0\log(n+1)$, with positive constants depending only on the source parameters. The argument refines the biclique counting method by charging the entropy of an adversarially chosen flattening to mutual information and a combinatorial obstruction energy. In the overlapping parameter range it strengthens the explicit soundness exponent previously established for this graph. We also establish an unconditional stability theorem for arbitrary sets $H$ of exactly $K$ distinct local assignments, with missing views and multiple answers: $f_*(H)\le34(n/\ell)q_{\mathrm{con}}(H)$ for every $1\le\ell\le n$. It gives a complementary additive near-clique soundness bound. We do not derive a new fine-grained hardness theorem.
作者备注：29 pages
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v1 · 2026-11-03 15:47 · 最新观测版本
Entropy-Weighted Soundness for Multi-Answer Birthday Graphs
The vertices of a Boolean birthday graph are partial assignments on $\ell$-subsets of $n$ variables. Although a satisfying assignment induces a $K$-clique for $K=\binom n\ell$, an arbitrary selection of $K$ vertices may repeat some views and omit others. This makes soundness substantially harder than under uniform view sampling. For bounded-occurrence, constant-gap Boolean ternary CSPs whose constraints each involve three distinct variables (allowing arbitrary Boolean predicates), we prove that the graph's maximum induced $K$-vertex density satisfies \[ ρ^*(K)\le 2\exp\!\left(-c\, \frac{\ell^3}{n^2\log(en/\ell)}\right) \] for $2\le\ell\le x_0n$ and $\ell\log(en/\ell)\ge C_0\log(n+1)$, with positive constants depending only on the source parameters. The argument refines the biclique counting method by charging the entropy of an adversarially chosen flattening to mutual information and a combinatorial obstruction energy. In the overlapping parameter range it strengthens the explicit soundness exponent previously established for this graph. We also establish an unconditional stability theorem for arbitrary sets $H$ of exactly $K$ distinct local assignments, with missing views and multiple answers: $f_*(H)\le34(n/\ell)q_{\mathrm{con}}(H)$ for every $1\le\ell\le n$. It gives a complementary additive near-clique soundness bound. We do not derive a new fine-grained hardness theorem.
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