We present new generalizations of Olson's theorem and of a consequence of
Alon's Combinatorial Nullstellensatz. These enable us to extend some of their
combinatorial applications with conditions modulo primes to conditions modulo
prime powers. We analyze computational search problems corresponding to these
kinds of combinatorial questions and we prove that the problem of finding
degree-constrained subgraphs modulo 2d such as 2d-divisible subgraphs and
the search problem corresponding to the Combinatorial Nullstellensatz over
F2 belong to the complexity class Polynomial Parity Argument (PPA)