  Many real discrete optimization problems (DOPs) are $NP$-hard and contain a
huge number of variables and/or constraints that make the models intractable
for currently available solvers. Large DOPs can be solved due to their special
tructure using decomposition approaches. An important example of decomposition
approaches is tree decomposition with local decomposition algorithms using the
special block matrix structure of constraints which can exploit sparsity in the
interaction graph of a discrete optimization problem. In this paper, discrete
optimization problems with a tree structural graph are solved by local
decomposition algorithms. Local decomposition algorithms generate a family of
related DO problems which have the same structure but differ in the right-hand
sides. Due to this fact, postoptimality techniques in DO are applied.
