A $(k,r)$-tuple is a word of length $r$ on an alphabet of size $k$. A graph is $(k,r)$-representable if we can assign a $(k,r)$-tuple to each vertex such that two vertices are connected iff the associated tuples agree on some component. We study the complexity of several graph problems on $(k,r)$-representable graphs, as a function of the parameters $k,r$; the problems under study are \textsc{Maximum Independent Set}, \textsc{Minimum Dominating Set} and \textsc{Maximum Clique}. In this framework, there are two classes of interest: the graphs representable with tuples of logarithmic length (\ie{} graphs $(k,r)$-representable with $r = O(k \log n)$), and the graphs representable with tuples of polynomial length (\ie{} graphs $(k,r)$-representable with $r = poly(n)$). In both cases, we show that the problems are computationally hard, though we obtain stronger hardness results in the second case. Our hardness results also allow us to derive optimality results for \textsc{Multidimensional Matching} and \textsc{Disjoint $r$-Subsets}.