2011/07/14 by Yacine Boufkhad, Boufkhad, Yacine, Olivier Dubois +1
Computer Science · Decision Sciences · #Advanced Graph Theory Research #Constraint Satisfaction and Optimization #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Scheduling and Timetabling Solutions #cs.DM
paper · pdf · doi:10.48550/arxiv.1107.2737
openalex publication_date 2011/07/14 · arxiv created 2011/09/25 · arxiv updated 2011/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The estimation of phase transitions in random boolean Constraint Satisfaction Problems (CSP) is based on two fundamental tools: the first and second moment methods. While the first moment method on the number of solutions permits to compute upper bounds on any boolean CSP, the second moment method used for computing lower bounds proves to be more tricky and in most cases gives only the trivial lower bound 0. In this paper, we define a subclass of boolean CSP covering the monotone versions of many known NP-Complete boolean CSPs. We give a method for computing non trivial lower bounds for any member of this subclass. This is achieved thanks to an application of the second moment method to some selected solutions called characteristic solutions that depend on the boolean CSP considered. We apply this method with a finer analysis to establish that the threshold rk (ratio : #constrains/#variables) of monotone 1-in-k-SAT is log k/k≤ rk≤log2k/k.