# Multivalued Dependency Inference Problem (Dependency Inference Problem)

## Description

The multivalued dependency inference problem is to find a cover for the set of multivalued dependencies that hold in a given relation.

A multivalued dependency (abbr. MVD), $g$, on a set of attributes $U$ is a statement $g: X \rightarrow \rightarrow Y$, where $X$ and $Y$ are subsets of $U$. Let $Z$ be the complement of the union of $X$ and $Y$ in $U$. A relation $R(U)$ obeys the MVD $g: X \rightarrow \rightarrow Y$ if for every $XZ$-value, $xz$, that appears in $R$, we have $Y_R(xz) = Y_R (x)$. In words, the MVD $g$ is valid in $R$ if the set of $Y$-values that appears in $R$ with a given $x$ appears with every combination of $x$ and $z$ in $R$. Thus, this set is a function of $x$ alone and does not depend on the $Z$-values that appear with $x$. Given $g: X \rightarrow \rightarrow Y$, we say that $g$ is a multivalued dependency from $X$ to $Y$ (in the set $U$). As we do for functional dependencies (FD's), here also we usually omit the name $g$ of the MVD and just write $X \rightarrow \rightarrow Y$.

## Parameters

$n$: number of attributes

$p$: number of tuples/rows/data points

## Table of Algorithms

Name Year Time Space Approximation Factor Model Reference
Räihä; Manilla 1992 $O(n^{2} {2}^n p log p)$ $O(n2^n)$? Exact Deterministic Time
Catriel Beeri Ronald Fagin John H. Howard 1977 $O({4}^n)$ Exact Deterministic Time