  We consider some computationally efficient and provably correct algorithms
with near-optimal sample-complexity for the problem of noisy non-adaptive group
testing. Group testing involves grouping arbitrary subsets of items into pools.
Each pool is then tested to identify the defective items, which are usually
assumed to be "sparse". We consider non-adaptive randomly pooling measurements,
where pools are selected randomly and independently of the test outcomes. We
also consider a model where noisy measurements allow for both some false
negative and some false positive test outcomes (and also allow for asymmetric
noise, and activation noise). We consider three classes of algorithms for the
group testing problem (we call them specifically the "Coupon Collector
Algorithm", the "Column Matching Algorithms", and the "LP Decoding Algorithms"
-- the last two classes of algorithms (versions of some of which had been
considered before in the literature) were inspired by corresponding algorithms
in the Compressive Sensing literature. The second and third of these algorithms
have several flavours, dealing separately with the noiseless and noisy
measurement scenarios. Our contribution is novel analysis to derive explicit
sample-complexity bounds -- with all constants expressly computed -- for these
algorithms as a function of the desired error probability; the noise
parameters; the number of items; and the size of the defective set (or an upper
bound on it). We also compare the bounds to information-theoretic lower bounds
for sample complexity based on Fano's inequality and show that the upper and
lower bounds are equal up to an explicitly computable universal constant factor
(independent of problem parameters).
