MAJORITY-3SAT (and Related Problems) in Polynomial Time

On the Satisfaction Probability of $k$-CNF Formulas

“…In sharp contrast to these well-established results, Akmal and Williams [1] recently showed that 3sat-prob >1/2 can be solved in polynomial time -in fact, they show this time bound for 3sat-prob >δ = {φ ∈ 3cnfs | σ(φ) > δ} for every δ > 0. Yet again in contrast, they also show that ksat-prob >1/2 is NP-complete for every k ≥ 4.…”

“…It was thus surprising that Akmal and Williams [1] were recently able to show that ksat-prob ≥δ ∈ P holds for all k and δ. As pointed out by Akmal and Williams, not only has the opposite generally been believed to hold, this has also been claimed repeatedly (page 1 of [1] lists no less than 15 different papers from the last 20 years that conjecture or even claim hardness of 3sat-prob ≥1/2 ). Just as surprising was the result of Akmal and Williams that while 4sat-prob ≥1/2 lies in P, the seemingly almost identical problem 4sat-prob >1/2 is NP-complete.…”

“…The algorithm of Akmal and Williams in [1] was the main inspiration for the results of the present paper and it shares a number of characteristics with Algorithm 1 (less with Algorithm 3 and none with Algorithm 2, though): Both algorithms search and then collapse sunflowers. However, without the Spectral Well-Ordering Theorem, one faces the problem that collapsing large sunflowers repeatedly could conceivably lower σ(φ) past δ.…”

“…However, without the Spectral Well-Ordering Theorem, one faces the problem that collapsing large sunflowers repeatedly could conceivably lower σ(φ) past δ. To show that this does not happen (without using the Spectral Well-Ordering Theorem) means that one has to redo all the arguments used in the proof of the Spectral Well-Ordering Theorem, but now with explicit parameters and constants and one has to intertwine the algorithmic and the underlying order-theoretic arguments in rather complex ways (just the analysis of the algorithm in [1] takes eleven pages in the main text plus two pages in the appendix).…”

“…From this perspective, Theorems 1.1 and 1.2 in [1] state models( ≥δ A ∀v 1 • • • ∀v k λ) ∈ LINTIME and models(∃A ≥δ B ∀v 1 • • • ∀v k λ) ∈ NP, respectively, for all k and quantifierfree λ. In contrast, the complexity of models( ≥ 1 2 A ≥ 1 2 B ∀v 1 • • • ∀v k λ) was not resolved by Akmal and Williams and the only explicitly named conjecture in [1] is that it lies in P. Phrased in terms of propositional logic, the conjecture states that there is a polynomialtime algorithm that, on input of a formula φ ∈ kcnfs with vars(φ) ⊆ X ∪ Y , finds out whether for a majority of the assignments β : X → {0, 1} the majority of extensions of β to β ′ : X ∪ Y → {0, 1} makes φ true (the "majority-of-majority problem, maj-maj-ksat").…”