Let G be a graph with vertex set V and a subset D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G). D is a total dominating set of G if every vertex in V is adjacent to a vertex in D. D is a restrained dominating set of G if every vertex in D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G) is adjacent to a vertex in D and another vertex in D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G). D is a total restrained dominating set if D is both a total dominating set and a restrained dominating set. The minimum cardinality of total dominating sets (resp. restrained dominating sets, total restrained dominating sets) of G is called the total domination number (resp. restrained domination number, total restrained domination number) of G, denoted by D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G) (resp. D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G), D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G)). The MINIMUM TOTAL RESTRAINED DOMINATION (MTRD) problem for a graph G is to find a total restrained dominating set of minimum cardinality of G. The TOTAL RESTRAINED DOMINATION DECISION (TRDD) problem is the decision version of the MTRD problem. In this paper, firstly, we show that the TRDD problem is NP-complete for undirected path graphs, circle graphs, S-CB graphs and C-CB graphs, respectively, and that, for a S-CB graph or C-CB graph with n vertices, the MTRD problem cannot be approximated within a factor of D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G) for any D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G) unless D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G). Secondly, for a graph G, we prove that the problem of deciding whether D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G) is NP-hard even when G is restricted to planar graphs with maximum degree at most 4, and that the problem of deciding whether D⊆VV\DV\Dγt(G)γr(G)γtr(G)(1-ϵ)lnnϵ>0NP⊆DTIME(nO(loglogn))γr(G)=γtr(G)γt(G)=γtr(G) is NP-hard even when G is restricted to planar bipartite graphs with maximum degree at most 5. Thirdly, we show that the MTRD problem is APX-complete for bipartite graphs with maximum degree at most 4. Finally, we design a linear-time algorithm for solving the MTRD problem for generalized series–parallel graphs.