In this thesis, we focus on distributed algorithms of the LOCAL model that run in constant time—that is, whose number of rounds is independent of the number of vertices—applied to the approximation of dominance problems within various classes of graphs. We provide MDS approximation algorithms for parameterized graph families, where the approximation factor does not depend on the parameter. First, we prove that given a LOCAL algorithm that produces a good approximation on planar graphs, it can be transformed into a LOCAL algorithm that produces a good approximation on graphs embeddable in a surface of Eulerian genus. Building on the algorithm of Heydt et al., we derive an approximation with a factor of 34+E for graphs of bounded genus. This result significantly improves upon the previous state of the art of 24g+O(1) established by Ami ri et al., as well as the factor of 91 H: obtained by Czygrinow et al. in the special case of orientable surfaces. 
We then generalize this result in two directions: first, by considering other graph problems studied in distributed computing, and second, by extending our results to classes of graphs beyond the bounded genus. We prove these results via a series of metatheorems concerning certain minimization problems. Next, we show that certain structured graphs (which exclude an H-minor with pathwidth at most 2) admit a deterministic distributed algorithm in f(H) rounds that computes a 50-approximation for the MDS problem. Although fast, approximate distributed algorithms for these problems were already known for graphs without minor H, all had an approximation factor dependent on H. A new key ingredient in the analysis of these various distributed algorithms is the use of the asymptotic dimension, a geometric concept introduced by Gromov in 1993.