Hereditary systems and greedy-type algorithms Original Research Article

We define a hereditary system on a finite set U as a partition of the family 2U of all subsets of U into disjoint families A and D satisfying (A∈A,A′⊆A)⇒A′∈A and (D∈D,D⊆D′)⇒D′∈D, respectively. The members of A are called independent sets, the sets D∈D are called dependent. We consider two important special cases of hereditary systems, matroids and comatroids, and study the structure of these objects. Two general combinatorial optimization problems on a hereditary system, the maximum independent set problem max{f(X) : X∈A} and the minimum dependent set problem min{f(X) : X∈D}, are considered. Jenkyns, Korte and Hausmann obtained a performance guarantee of the greedy heuristic (‘best in’) for the maximum independent set problem. We present a greedy-type approximation algorithm for solving the minimum dependent set problem, the steepest descent heuristic (‘worst out’), study interconnections between the above-mentioned problems and derive performance guarantees of the steepest descent algorithm. Finally, we apply our results to obtain performance guarantees of the steepest descent algorithm for some known special combinatorial minimization problems.