Discussion of “On the Set Representation and the Set Covering Problem”

Abstract

Murty points out that the set representation problem and the set covering problems are equivalent. A very easy way to see this is to construct a bipartite graph for each problem. For the representation problem, the nodes in one part are identified with elements of the set Γ and nodes in the other part with subsets in the class C. There exists an arc (i, j) iff element i belongs to subset c. For the covering problem, the nodes in one part are identified with subsets ξ_i in the class θ and nodes in the other part with elements of the underlying set. There exists an arc (i, j) iff element j belongs to subset ξ_i In both cases, a feasible solution (i.e., a represent or a cover) consists of a subset S of nodes in the first-mentioned part such that each node in the other part is adjacent to at least one node in S. Minimal solutions and minimum cardinality solutions have obvious interpretations.