  In the usual trace reconstruction problem, the goal is to exactly reconstruct
an unknown string of length $n$ after it passes through a deletion channel many
times independently, producing a set of traces (i.e., random subsequences of
the string). We consider the relaxed problem of approximate reconstruction.
Here, the goal is to output a string that is close to the original one in edit
distance while using much fewer traces than is needed for exact reconstruction.
We present several algorithms that can approximately reconstruct strings that
belong to certain classes, where the estimate is within $n/\mathrm{polylog}(n)$
edit distance, and where we only use $\mathrm{polylog}(n)$ traces (or sometimes
just a single trace). These classes contain strings that require a linear
number of traces for exact reconstruction and which are quite different from a
typical random string. From a technical point of view, our algorithms
approximately reconstruct consecutive substrings of the unknown string by
aligning dense regions of traces and using a run of a suitable length to
approximate each region. To complement our algorithms, we present a general
black-box lower bound for approximate reconstruction, building on a lower bound
for distinguishing between two candidate input strings in the worst case. In
particular, this shows that approximating to within $n^{1/3 - \delta}$ edit
distance requires $n^{1 + 3\delta/2}/\mathrm{polylog}(n)$ traces for $0< \delta
< 1/3$ in the worst case.
