Approximate Keys and Functional Dependencies in Incomplete Databases With Limited Domains-Algorithmic Perspective

A possible world of an incomplete database table is obtained by imputing values from the attributes (infinite) domain to the place of \texttt{NULL} s. A table satisfies a possible key or possible functional dependency constraint if there exists a possible world of the table that satisfies the given key or functional dependency constraint. A certain key or functional dependency is satisfied by a table if all of its possible worlds satisfy the constraint. Recently, an intermediate concept was introduced. A strongly possible key or functional dependency is satisfied by a table if there exists a strongly possible world that satisfies the key or functional dependency. A strongly possible world is obtained by imputing values from the active domain of the attributes, that is from the values appearing in the table. In the present paper, we study approximation measures of strongly possible keys and FDs. Measure $g_3$ is the ratio of the minimum number of tuples to be removed in order that the remaining table satisfies the constraint. We introduce a new measure $g_5$, the ratio of the minimum number of tuples to be added to the table so the result satisfies the constraint. $g_5$ is meaningful because the addition of tuples may extend the active domains. We prove that if $g_5$ can be defined for a table and a constraint, then the $g_3$ value is always an upper bound of the $g_5$ value. However, the two measures are independent of each other in the sense that for any rational number $0\le\frac{p}{q}<1$ there are tables of an arbitrarily large number of rows and a constant number of columns that satisfy $g_3-g_5=\frac{p}{q}$. A possible world is obtained usually by adding many new values not occurring in the table before. The measure $g_5$ measures the smallest possible distortion of the active domains. We study complexity of determining these approximate measures.