SAT Distributions with planted assignments and phase transitions between decision and optimization problems

• Authors: Tassos Dimitriou
• Venue: Discrete Applied Mathematics - Special issue: Typical case complexity and phase transitions
• Year: 2005

We present a generator for weighted instances of MAX k-SAT in which every clause has a weight associated with it and the goal is to maximize the total weight of satisfied clauses. Our generator produces formulas whose hardness can be finely tuned by two parameters p and δ that control the weights of the clauses. Under the right choice of these parameters an easy-hard-easy pattern in the search complexity emerges which is similar to the patterns observed for traditional SAT distributions.What is remarkable, however, is that the generated distributions seem to lie in the middle ground between decision and optimization problems. Increasing the value of p from 0 to 1 has the effect of changing the shape of the computational cost from an easy-hard-easy pattern which is typical of decision problems to an easy-hard pattern which is typical of optimization problems. Thus our distributions seem to bridge the gap between decision and optimization versions of SAT.Furthermore, we demonstrate that these phase transitions are related to sudden changes to a quantity similar to the backbone of a SAT formula. In our model not only we know how the optimal solution looks like (because we plant it in advance) but we also give evidence that it is unique. Thus our generator comes with an indication of optimality of the planted assignment which is basically the structural property that is related to the phase transition phenomena observed.