Clutter nonidealness

Several key results for set packing problems do not seem to be easily or even possibly transferable to set covering problems, although the symmetry between them. The goal of this paper is to introduce a nonidealness index by transferring the ideas used for the imperfection index defined by Gerke and McDiarmid Graph imperfection, J. Combin. Theory Ser. B 83 (2001) 58-78]. We found a relationship between the two indices and the strength of facets defined in M. Goemans, Worst-case comparison of valid inequalities for the TSP, mathematical programming, in: Fifth Integer Programming and Combinatorial Optimization Conference, Lecture Notes in Computer Science, vol. 1084, Vancouver, Canada, 1996, pp. 415-429; M. Goemans, L.A. Hall, The strongest facets of the acyclic subgraph polytope are unknown, in: Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 1084, Springer, Berlin, 1996, pp. 415-429]. We prove that a clutter is as nonideal as its blocker and find some other properties that could be transferred from the imperfection index to the nonidealness index. Finally, we analyze the behavior of the nonidealness index under some clutter operations.