Summary: Let \(G=(V,E)\) be a graph. A double Roman dominating function (DRDF) of \(G\) is a function \(f:V\to \{0,1,2,3\}\) such that, for each \(v\in V\) with \(f(v)=0\), there is a vertex \(u\) adjacent to \(v\) with \(f(u)=3\) or there are vertices \(x\) and \(y\) adjacent to \(v\) such that \(f(x)=f(y)=2\) and for each \(v\in V\) with \(f(v)=1\), there is a vertex \(u\) adjacent to \(v\) with \(f(u)>1\). The weight of a DRDF \(f\) is \(f (V) =\sum_{ v\in V} f (v)\). Let \(n\) and \(k\) be integers such that \(3\leq 2k+ 1 \leq n\). The generalized Petersen graph \(GP (n, k)=(V,E)\) is the graph with \(V=\{u_1, u_2,\ldots, u_n\}\cup\{v_1, v_2,\ldots, v_n\}\) and \(E=\{u_iu_{i+1}, u_iv_i, v_iv_{i+k}: 1 \leq i \leq n\}\), where addition is taken modulo \(n\). In this paper, we firstly prove that the decision problem associated with double Roman domination is NP-complete even restricted to planar bipartite graphs with maximum degree at most 4. Next, we give a dynamic programming algorithm for computing a minimum DRDF (i.e., a DRDF with minimum weight along all DRDFs) of \(GP(n,k)\) in \(O(n81^k)\) time and space and so a minimum DRDF of \(GP(n,O(1))\) can be computed in \(O( n)\) time and space.