Generating Weighted MAX-2-SAT Instances of Tunable Difficulty with Frustrated Loops

Many optimization problems can be cast into the maximum satisfiability (MAX-SAT) form, and many solvers have been developed for tackling such problems. To evaluate the performance of a MAX-SAT solver, it is convenient to generate difficult MAX-SAT instances with solutions known in advance. Here, we propose a method of generating weighted MAX-2-SAT instances inspired by the frustrated-loop algorithm used by the quantum annealing community to generate Ising spin-glass instances with nearest-neighbor coupling. Our algorithm is extended to instances whose underlying coupling graph is general, though we focus here on the case of bipartite coupling, with the associated energy being the restricted Boltzmann machine (RBM) energy. It is shown that any MAX-2-SAT problem can be reduced to the problem of minimizing an RBM energy over the nodal values. The algorithm is designed such that the difficulty of the generated instances can be tuned through a central parameter known as the frustration index. Two versions of the algorithm are presented: the random- and structured-loop algorithms. For the random-loop algorithm, we provide a thorough theoretical and empirical analysis on its mathematical properties from the perspective of frustration, and observe empirically, using simulated annealing, a double phase transition behavior in the difficulty scaling behavior driven by the frustration index. For the structured-loop algorithm, we show that it offers an improvement in difficulty of the generated instances over the random-loop algorithm, with the improvement factor scaling super-exponentially with respect to the frustration index for instances at high loop density. At the end of the paper, we provide a brief discussion of the relevance of this work to the pre-training of RBMs.