  We introduce a reformulation technique that converts a many-set feasibility
problem into an equivalent two-set problem. This technique involves
reformulating the original feasibility problem by replacing a pair of its
constraint sets with their intersection, before applying Pierra's classical
product space reformulation. The step of combining the two constraint sets
reduces the dimension of the product spaces. We refer to this as the constraint
reduction reformulation and use it to obtain constraint-reduced variants of
well-known projection algorithms such as the Douglas--Rachford algorithm and
the method of alternating projections, among others. We prove global
convergence of constraint-reduced algorithms in the presence of convexity and
local convergence in a nonconvex setting. In order to analyse convergence of
the constraint-reduced Douglas--Rachford method, we generalize a classical
result which guarantees that the composition of two projectors onto subspaces
is a projector onto their intersection. Finally, we apply the
constraint-reduced versions of Douglas--Rachford and alternating projections to
solve the wavelet feasibility problems, and then compare their performance with
their usual product variants.
