title: Satisfiability-unsatisfiability transition in the adversarial satisfiability problem.
abstract: Adversarial satisfiability (AdSAT) is a generalization of the satisfiability (SAT) problem in which two players try to make a Boolean formula true (resp. false) by controlling their respective sets of variables. AdSAT belongs to a higher complexity class in the polynomial hierarchy than SAT, and therefore the nature of the critical region and the transition are not easily parallel to those of SAT and worthy of independent study. AdSAT also provides an upper bound for the transition threshold of the quantum satisfiability problem (QSAT). We present a complete algorithm for AdSAT, show that 2-AdSAT is in P, and then study two stochastic algorithms (simulated annealing and its improved variant) and compare their performances in detail for 3-AdSAT. Varying the density of clauses α we claim that there is a sharp SAT-UNSAT transition at a critical value whose upper bound is αc≲1.5, suggesting a much stricter upper bound for the QSAT transition than those previously found.
authors.affiliation: Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy.
authors.forename: Marco
authors.initials: M
authors.lastname: Bardoscia
authors.affiliation: Dept. of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 845 11 Bratislava, Slovakia.
authors.forename: Daniel
authors.initials: D
authors.lastname: Nagaj
authors.affiliation: Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy and INFN, Section of Trieste, Strada Costiera 11, 34151 Trieste, Italy.
authors.forename: Antonello
authors.initials: A
authors.lastname: Scardicchio