For a fixed positive integer , a -tuple total dominating set of a graph is a set such that every vertex of is adjacent to at least vertices in . The -tuple total domination problem is to determine a minimum -tuple total dominating set of . This paper studies -tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for the -tuple total domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph, which include trees, block graphs, cacti and block-cactus graphs. We also establish NP-hardness of the -tuple total domination problem in undirected path graphs.