From 6326886030f8ba753116744b581e6dd84c00e76d Mon Sep 17 00:00:00 2001 From: Samuel Giuseppe Schlesinger Date: Fri, 30 Oct 2026 17:21:10 -0400 Subject: [PATCH] doc(pcp): clarify the perfect-completeness hardness roadmap Keep the satisfiable-versus-7/8+epsilon MAX-E3SAT endpoint tied to small-soundness Label Cover and Hastad separate perfect-completeness construction. The MAX-E3LIN/XOR route supplies the weaker general approximation result, not this prerequisite chain. Checked the primary JACM Theorem 6.1/6.5 source mapping and whitespace. Documentation only; Lean sources, selected abstracts and rendering are unaffected. Follows the nonblocking source-scope finding in review 5718563432 of PR #1800; refs #886. AI assistance: Codex, worker-codex-pcp-20261030-65g7glwh. --- ROADMAP.md | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/ROADMAP.md b/ROADMAP.md index fe1c32dcd3..c9fb069709 100644 --- a/ROADMAP.md +++ b/ROADMAP.md @@ -323,9 +323,8 @@ DLOGTIME-uniform AC⁰ under explicit conventions. - **Håstad's optimal MAX-E3SAT hardness** ([#886]): for `0 < ε < 1/8`, NP-hardness of distinguishing satisfiable E3-CNF formulas from those where every assignment satisfies at most - `7/8 + ε` of the clauses. This needs the reduction to MAX-E3LIN and, through the 4-clause XOR - gadget, to MAX-E3SAT; Raz–Holenstein parallel repetition for small-soundness Label Cover; and - the perfect-completeness reduction. + `7/8 + ε` of the clauses. This needs Raz–Holenstein parallel repetition for small-soundness + Label Cover and Håstad's separate perfect-completeness construction for MAX-E3SAT. - The FGLSS clique-hardness reduction and its amplification, retaining the exact gap and reduction runtime. - Gap and approximation-preserving reductions, APX/PTAS/FPTAS, inapproximability consequences of