A Class of Polynomially Solvable Set-Covering Problems

Abstract

A clutterL is a collection of m subsets of a ground set $E( L ) = \{ x_1 , \cdots ,x_n \}$ with the property that for every pair $A_i ,A_j \in L,A_i $ is neither contained nor contains $A_j $. A transversal of L is a subset of $E( L )$ having at least one element in common with each member of L.The problem of finding the minimum weight transversal of a clutter L is equivalent to the well-known set-covering problem.In this paper the class of ideal clutters that properly contains the class of clutters the members of which are the bases of a matroid (matroidal clutters) is introduced.An ideal clutter L has the property that the number of its minimal transversals is bounded by a polynomial in m and n. The properties of ideal clutters are described, and two polynomial algorithms for recognizing them and finding their minimum weight transversal are presented.