A general question on NP class problem
In short: no, it doesn't follow from the claims you mention that this algorithm is a verifier in a meaningful use of the term.

The paper states:


  Our attempt here is to obtain a polynomial-time algorithm for
  computing these tables that is sufficient but not necessary,
  in the following sense. Our algorithm will not be able to
  generate the scheduling tables for all TT-schedulable instances;
  however, if the algorithm does generate scheduling tables for
  an instance, then these tables are correct.


So, they claim to have a polynomial time algorithm that can provide a solution for a subset of all problem instances, but there are some cases for which this algorithm fails to provide a correct solution. Furthermore, they claim elsewhere that there is some characterization of this subset.

By contrast, a polynomial time verifier for a certain problem $X$ is an algorithm, that given an problem instance $I$ of $X$ and a certificate $C$, can prove whether the instance is a YES-instance or not in polynomial time. Usually, the certificate corresponds to some 'solution' that the verifier needs to verify correctness for.

So, the algorithm in this paper can be called a polynomial time verifier algorithm for the subset it solves in polynomial time only, as a polynomial algorithm that solves a problem is also a verifier. But, the claim that this algorithm is a verifier in this sense isn't useful, as it is a strictly weaker property (at least, if $P\neq NP$ ) than being an actual algorithm for the solution.