  Consider a query-based data acquisition problem that aims to recover the
values of $k$ binary variables from parity (XOR) measurements of chosen subsets
of the variables. Assume the response model where only a randomly selected
subset of the measurements is received. We propose a method for designing a
sequence of queries so that the variables can be identified with high
probability using as few ($n$) measurements as possible. We define the query
difficulty $\bar{d}$ as the average size of the query subsets and the sample
complexity $n$ as the minimum number of measurements required to attain a given
recovery accuracy. We obtain fundamental trade-offs between recovery accuracy,
query difficulty, and sample complexity. In particular, the necessary and
sufficient sample complexity required for recovering all $k$ variables with
high probability is $n = c_0 \max\{k, (k \log k)/\bar{d}\}$ and the sample
complexity for recovering a fixed proportion $(1-\delta)k$ of the variables for
$\delta=o(1)$ is $n = c_1\max\{k, (k \log(1/\delta))/\bar{d}\}$, where $c_0,
c_1>0$.
