A greedy algorithm for maximizing a linear objective function Original Research Article

It is well known that the problem of finding a maximum-weight base of matroid can be solved by a greedy algorithm. The aim of this paper is to extend this result for some more general problems. One of these problems is maximizing a linear objective function over a family I of integer nonnegative vectors. We establish a number of necessary and sufficient conditions for the greedy algorithm to find an optimal solution in the general and some special cases. For the Boolean case, Goecke et al. (in: Combinatorial Optimization, Lecture Notes in Mathematics, Vol. 1403, Springer, Berlin, 1986, p. 1986) proved that the algorithm works if and only if the so-called strong exchange property holds. We extend this result to a class of non-Boolean vector systems.