  This paper introduces a family of valid inequalities, that we term
consistency cuts, to be applied to a Dantzig-Wolfe reformulation (or
decomposition) with linking variables. We prove that these cuts ensure an
integer solution to the corresponding Dantzig-Wolfe relaxation when certain
criteria to the structure of the decomposition are met. We implement the cuts
and use them to solve a commonly used test set of 200 instances of the temporal
knapsack problem. We assess the performance with and without the cuts and
compare further to CPLEX and other solution methods that have historically been
used to solve the test set. By separating consistency cuts we show that we can
obtain optimal integer solutions much faster than the other methods and even
solve the remaining unsolved problems in the test set. We also perform a second
test on instances from the MIPLIB 2017 online library of mixed-integer
programs, showing the potential of the cuts on a wider range of problems.
