The question asks whether a randomized DPLL algorithm (splitting on random variables) can find a satisfying assignment for "easy" CNF formulas (each clause has exactly ( t ) distinct literals, and each clause shares variables with fewer than ( 2^t/e ) others) in polynomial expected time.

Key Analysis:

1. Easy Formulas and LLL: Easy formulas are satisfiable by the Lovász Local Lemma (...

Short answer: No. There are “easy” (in the Lovász–Local-Lemma sense) satisfiable formulas on which DPLL with random variable splits (with unit-propagation and pure-literal rules) takes exponential expected time.

Why the LLL condition does not help DPLL

• The LLL condition (each t-clause conflicts with fewer than 2^t/e other clauses) guarantees existence of a satisfying assignment and, via the Mos...

This is an excellent question about the relationship between two fundamental approaches to SAT solving: the algorithmic version of the Lovász Local Lemma (LLL) and the DPLL algorithm.

The Answer

No, it is not known whether DPLL with random variable selection finds satisfying assignments for all "easy" formulas (those satisfying the LLL condition) in polynomial expected time. In fact, ther...

At the moment the question is open – we do not know a proof that a DPLL–type procedure with a random branching rule always finishes in expected polynomial time on every CNF that satisfies the (symmetric) Lovász Local Lemma condition, and no counter–example is known either.

What is known.

1. Moser–Tardos (2009) showed that the simple “resample-an-unsatisfied-clause’’ algorithm finds a satisf...

First, let me restate the problem in more “proof‐system” language. A classic fact is that

1. If you have an LLL‐easy CNF—i.e. every clause has size exactly (t), and each clause shares variables with at most (\tfrac{2^t}{e}) other clauses—then the Lovász Local Lemma shows a satisfying assignment exists.

2. Moser–Tardos then give a randomized local‐resampling procedure (“pick any vi...

Excellent question. It gets to the heart of the difference between constructive theoretical proofs and the behavior of practical algorithms.

The answer is very likely no. A standard DPLL algorithm, even with random variable selection, is not expected to solve all "easy" (in the LLL sense) formulas in polynomial time.

Here's a breakdown of the reasoning, from the high-level intuition to a ske...

Short answer
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Nobody knows. At the moment there is neither

• a proof that a (pure) DPLL procedure that always branches on a randomly chosen variable finishes in polynomial expected time on every instance that satisfies the Lovász-Local-Lemma (LLL) condition, nor

• an instance that satisfies this LLL condition for which the same DPLL procedure can be shown to need super-p...

<think> We are given a CNF formula $\varphi$ where each clause has exactly $t$ literals, and all literals in a clause correspond to different variables. Moreover, we are told that the formula is "easy", meaning that every clause has fewer than $2^t / e$ clauses that share variables with it. This condition comes from the Lovász Local Lemma (LLL) and guarantees that the formula is satisfiable.

The ...

DPLL and Lovász Local Lemma: Runtime Analysis

This question asks whether a DPLL algorithm that randomly selects variables for branching can find satisfying assignments for "easy" formulas (those satisfying the Lovász Local Lemma conditions) in polynomial expected time.

The Key Distinction

There's a fundamental difference between the Moser-Tardos algorithm and DPLL with random variable sele...