On the Minimal Constraint Satisfaction Problem: Complexity and Generation
- 754 Downloads
The Minimal Constraint Satisfaction Problem, or Minimal CSP for short, arises in a number of real-world applications, most notably in constraint-based product configuration. Despite its very permissive structure, it is NP-hard, even when bounding the size of the domains by \(d\ge 9\). Yet very little is known about the Minimal CSP beyond that. Our contribution through this paper is twofold. Firstly, we generalize the complexity result to any value of d. We prove that the Minimal CSP remains NP-hard for \(d\ge 3\), as well as for \(d=2\) if the arity k of the instances is strictly greater than 2. Our complexity result can be seen as providing a dichotomy theorem for the Minimal CSP. Secondly, we build a generator that can create Minimal CSP instances of any size, using the constrainedness as a parameter. Our generator can be used to study behaviors that are typical of NP-hard problems, such as the presence of a phase transition, in the case of the Minimal CSP.
KeywordsPermission Structure Desired Tightness Average Tightness Binary Boolean Tern Instances
This publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289.
- 1.Achlioptas, D., Gomes, C., Kautz, H., Selman, B.: Generating satisfiable problem instances. In: Kautz, H.A., Porter, B.W. (eds.) Proceedings of AAAI, pp. 256–261. AAAI Press/The MIT Press (2000)Google Scholar
- 3.Clark, D.A., Frank, J., Gent, I.P., MacIntyre, E., Tomov, N., Walsh, T.: Local search and the number of solutions. In: Freuder, Eugene C. (ed.) CP 1996. LNCS, vol. 1118. Springer, Heidelberg (1996) Google Scholar
- 5.Gent, I.P., MacIntyre, E., Prosser, P., Walsh, T.: The constrainedness of search. In: Clancey, W.J., Weld, D.S. (eds.) Proceedings of AAAI, pp. 246–252. AAAI Press/The MIT Press (1996)Google Scholar
- 8.Junker, U.: Configuration. In: Handbook of Constraint Programming. Foundations of Artificial Intelligence, pp. 837–873. Elsevier (2006)Google Scholar
- 9.Schaefer, T.J.: The complexity of satisfiability problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, 1–3 May 1978, San Diego, pp. 216–226. ACM (1978)Google Scholar