On Random Ordering Constraints

Authors:
Andreas Goerdt
Affiliations:
Fakultät für Informatik, Technische Universität Chemnitz, Chemnitz, Germany 09107
Venue:
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Year:
2009

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Abstract

Ordering constraints are analogous to instances of the satisfiability problem in conjunctive normalform, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering iff the relative ordering of its variables obeys the constraint. The naturally arising satisfiability problems are NP-complete for many types of constraints. The present paper seems to be one of the first looking at random ordering constraints. Experimental evidence suggests threshold phenomena as in the case of random k -SAT instances. We prove first that random instances of the cyclic ordering and betweenness constraint have a sharp threshold for unsatisfiability. Second, random instances of the cyclic ordering constraint are satisfiable with high probability if the number of clauses is $\le 1 \times\,\, \sharp$variables.