Given a finite set E and a real valued function f on P(E) (the power set of E) the optimal subset problem (P) is to find S ⊂ E maximizing f over P(E). Many combinatorial optimization problems can be formulated in these terms. Here, a family of approximate solution methods is studied : the greedy algorithms.
After having described the standard greedy algorithm (SG) it is shown that, under certain assumptions (namely : submodularity of f) the computational complexity of (SG) can often be significantly reduced, thus leading to an accelerated greedy algorithm (AG). This allows treatment of large scale combinatorial problems of the (P) type. The accelerated greedy algorithm is shown to be optimal (interms of computational complexity) over a wide class of algorithms, and the submodularity assumption is used to derive bounds on the difference between the greedy solution and the optimum solution.