The width of a problem is the width of its minimal-width decomposition. While decompositions of fixed width can be used to efficiently solve a problem, a bound on the width of instances does necessarily produce a tractable structural restriction. Indeed, a fixed width problem has a decomposition of fixed width, but finding it may not be polynomial. In order for a problem of fixed width being efficiently solved by decomposition, one of its decompositions of low width has to be found efficiently. For this reason, decomposition methods and their associated width are defined in such a way not only solving the problem given a fixed-width decomposition of it is polynomial-time, but also finding a fixed width decomposition of a fixed-width problem is polynomial-time.