Abstract
Fulkerson et al. have given two examples of set covering problems that are empirically difficult to solve. They arise from Steiner triple systems and the larger problem, which has a constraint matrix of size 330 × 45 has only recently been solved. In this note, we show that the Steiner triple systems do indeed give rise to a series of problems that are probably hard to solve by implicit enumeration. The main result is that for ann variable problem, branch and bound algorithms using a linear programming relaxation, and/or elimination by dominance require the examination of a super-polynomial number of partial solutions
Similar content being viewed by others
References
V. Chvátal, “Hard knapsack problems”,Journal of Operations Research, to appear.
V. Chvátal, “A greedy heuristic for the set covering problem”, Publication No. 284, Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, Montréal (1978).
V. Chvátal, “Determining the stability number of a graph”,SIAM Journal on Computing 6 (1977) 643–662.
S.A. Cook and R.A. Reckhow, “On the Lengths of Proofs in the Propositional Calculus”,Proceedings of 6th Annual ACM Symposium on Theory of Computing (1974) 135–148.
R. Fulkerson, G. Nemhauser and L. Trotter, “Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems”,Mathematical Programming Study 2 (1974) 72–81.
R. Garfinkel and G. Nemhauser,Integer programming (Wiley, New York, 1969).
A. Geoffrion, “An improved implicit enumeration approach for integer programming”,Journal of Operations Research 17 (1969) 437–454.
M. Hall Jr.,Combinatorial theory (Blaisdell Company, Waltham, MA, 1967).
C.E. Lemke, H.M. Salkin and K. Spielberg, “Set covering by single branch enumeration with linear programming subproblems”,Journal of Operations Research 19 (1971) 998–1022.
C. McDiarmid, “Determining the chromatic number of a graph”,SIAM Journal on Computing 8 (1979) 1–14.
Author information
Authors and Affiliations
Additional information
This paper was written while the author was a CORE Fellow at the Université de Louvain, Louvain-la-Neuve, Belgium.
Rights and permissions
About this article
Cite this article
Avis, D. A note on some computationally difficult set covering problems. Mathematical Programming 18, 138–145 (1980). https://doi.org/10.1007/BF01588309
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01588309