Karp-Luby Sampling

Synonyms

Definition

Karp-Luby (KL) sampling [1, 2] is a Monte Carlo algorithm that is used to estimate the number of truth assignments to a set of literals x = {x₁, …, x_k} that satisfy a logical expression L[x] having “disjunctive normal form” (DNF). That is, L[x] has the form C₁ ∨ C₂ ∨⋯ ∨ C_m, where each clause C_i is a conjunction of the form $C_i=y_{i_1}\wedge y_{i_2}\wedge \cdots \wedge y_{i_l}$ with i₁, i₂, …, i_l being distinct elements of {1, 2, …, k} and $y_{i_j}=x_{i_j}$ or $\neg x_{i_j}$ for 1 ≤ j ≤ l. A straightforward modification of the technique can be used to estimate P(L[X] = 1), where X = {X₁, X₂, …, X_k} is a set of mutually independent boolean random variables. This latter use of KL sampling plays a key role when querying uncertain data. Exact computation is #P-complex in the data size both for the original counting problem problem and for computation of p = P(L[X] = 1), which is why approximate methods are needed. A naive approach that simply generates n...