Exact thresholds for DPLL on random XOR-SAT and NP-complete extensions of XOR-SAT
Theoretical Computer Science
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Despite much work over the previous decade, the Satisfiability Threshold Conjecture remains open. Random k-SAT, for constant k ≥ 3, is just one family of a large number of constraint satisfaction problems that are conjectured to have exact satisfiability thresholds, but for which the existence and location of these thresholds has yet to be proven. Of those problems for which we are able to prove an exact satisfiability threshold, each seems to be fundamentally different than random 3-SAT.This thesis defines a new family of constraint satisfaction problems with constant size constraints and domains and which contains problems that are NP-complete and a.s. have exponential resolution complexity. All four of these properties hold for k-SAT, k ≥ 3, and the exact satisfiability threshold is not known for any constraint satisfaction problem that has all of these properties. For each problem in the family defined in this thesis, we determine a value c such that c is an exact satisfiability threshold if a certain multi-variable function has a unique maximum at a given point in a bounded domain. We also give numerical evidence that this latter condition holds. In addition to studying the satisfiability threshold, this thesis finds exact thresholds for the efficient behavior of DPLL using the unit clause heuristic and a variation of the generalized unit clause heuristic, and this thesis proves an analog of a conjecture on the satisfiability of (2 + p)-SAT.Besides having similar properties as k-SAT, this new family of constraint satisfaction problems is interesting to study in its own right because it generalizes the XOR-SAT problem and it has close ties to quasigroups.