bounds in centralized and distributed If we know some lower bound of the solution of a problem in centralized setting, what can we say about the lower bound in a distributed setting?
I'm not aware of a general rule to convert centralized bounds to the distributed setting. In the distributed setting, local computation is given for free but the difficulty lies in breaking the symmetry - a node needs to accumulate enough information about it's neighborhood to make a decision about being part of the output set, e.g. being in the MIS. (Taking this a step further, we can think of an $r$-round algorithm as a function mapping $r$-neighborhoods to a yes/no decision.)

For example, for certain $n$-node graphs with max degree $\Delta$ there is an $\Omega(\sqrt{\log n}+\log \Delta)$ lower bound [2] known on the number of distributed rounds, that holds for (the distributed version of) computing a constant approximation of a minimum vertex cover or of a minimum dominating set. 

While the above problems are known to be NP-complete, the same lower bound also holds for problems for which there are polynomial time centralized algorithms like computing a maximal independent set or finding a maximal matching. 

For most of these problems the lower bound is almost tight; there are distributed algorithms that run in polylogarithmic rounds (see [2] and references therein).

On the other hand, some NP complete problems even allow constant time solutions in the distributed setting! In [1] there's an $O(1)$ round distributed algorithm for $O(n^{1/2+\varepsilon} \chi)$-coloring, which itself is NP complete.

[1] Leonid Barenboim. On the Locality of Some NP-Complete Problems
http://arxiv.org/abs/1204.6675

[2] Fabian Kuhn, Thomas Moscibroda, Roger Wattenhofer. Local Computation: Lower and Upper Bounds.
http://arxiv.org/pdf/1204.6675v1.pdf