For a simple, undirected graph G ¼ ðV, EÞ, a Roman {2}-dominating function (R2DF) f : V ! f0, 1, 2g has the property that for every vertex v 2 V with f(v) ¼ 0, either there exists a vertex u 2 NðvÞ, with f(u) ¼ 2, or at least two vertices x, y 2 NðvÞ with f ðxÞ ¼ f ðyÞ ¼ 1: The weight of an R2DF is the sum f ðVÞ ¼ P v2V f ðvÞ: The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by c fR2g ðGÞ: A double Roman dominating function (DRDF) on G is a function f : V ! f0, 1, 2, 3g such that for every vertex v 2 V if f(v) ¼ 0, then v has at least two neighbors x, y 2 NðvÞ with f ðxÞ ¼ f ðyÞ ¼ 2 or one neighbor w with f(w) ¼ 3, and if f(v) ¼ 1, then v must have at least one neighbor w with f ðwÞ ! 2: The weight of a DRDF is the value f ðVÞ ¼ P v2V f ðvÞ: The minimum weight of a DRDF is called the double Roman domination number and is denoted by c dR ðGÞ: Given an graph G and a positive integer k, the R2DP (DRDP) problem is to check whether G has an R2DF (DRDF) of weight at most k. In this article, we first show that the R2DP problem is NP-complete for star convex bipartite graphs, comb convex bipartite graphs and bisplit graphs. We also show that the DRDP problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Next, we show that c fR2g ðGÞ, and c dR ðGÞ are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a 2ð1 þ ln ðD þ 1ÞÞ-approximation algorithm for the minimum Roman {2}-domination problem and 3ð1 þ ln ðD þ 1ÞÞ-approximation algorithm for the minimum double Roman domination problem, where D is the maximum degree of G.