  A distance-approximation algorithm for a graph property $\mathcal{P}$ in the
adjacency-matrix model is given an approximation parameter $\epsilon \in (0,1)$
and query access to the adjacency matrix of a graph $G=(V,E)$. It is required
to output an estimate of the \emph{distance} between $G$ and the closest graph
$G'=(V,E')$ that satisfies $\mathcal{P}$, where the distance between graphs is
the size of the symmetric difference between their edge sets, normalized by
$|V|^2$. In this work we introduce property covers, as a framework for using
distance-approximation algorithms for "simple" properties to design
distance-approximation. Applying this framework we present
distance-approximation algorithms with $poly(1/\epsilon)$ query complexity for
induced $P_3$-freeness, induced $P_4$-freeness, and Chordality. For induced
$C_4$-freeness our algorithm has query complexity $exp(poly(1/\epsilon))$.
These complexities essentially match the corresponding known results for
testing these properties and provide an exponential improvement on previously
known results.
