Scaling and Probabilistic Smoothing (SAPS)

The SAPS algorithm is a Dynamic Local Search (DLS) algorithm conceptually closely related to the Exponentiated Sub-Gradient (ESG) algorithm developed by Schuurmans, Southey and Holte [3]. When introducing SAPS, our major contributions were a reduction in the algorithmic complexity as compared to the ESG algorithm and a new perspective on how the two algorithms were behaving. The SAPS algorithm is described in detail in our paper [2] and Figure 1 contains a pseudo-code representation that accurately reflects how the SAPS algorithm has been implemented in practice. Similar to most DLS algorithms, SAPS assigns a clause penalty clp to each clause, and the search evaluation function of SAPS is the sum of the clause penalties of unsatisfied clauses. The core search procedure is a greedy descent without sideways steps. Whenever a local minimum occurs (no step improvement in the evaluation function greater than SAPSthresh is possible) a random walk step occurs with probability wp. Otherwise, a scaling step occurs, where the penalties for unsatisfied clauses are multiplied by the scaling factor α (i.e. clp := α · clp). After a scaling step, a smoothing step occurs with probability Psmooth. In a smoothing step, all penalties are adjusted according to the mean penalty value clp and the smoothing factor ρ (i.e. clp := clp + (1 − ρ) · clp). Along with the SAPS algorithm, we also developed a reactive variant (RSAPS) [2] that reactively changes the smoothing parameter ρ during the search process whenever search stagnation is detected, using the same adaptive mechanism as Adaptive Novelty [1]. More recently we have developed a de-randomised variant of SAPS called SAPS/NR [6], which eliminates all sources of random decisions throughout the search (breaking ties deterministically, performing periodic smoothing, and no random walk steps) and which relies upon the initial random variable assignment as the only source of randomness. In our experiments, we have found that SAPS, RSAPS and SAPS/NR are amongst the state-of-theart SLS SAT solvers, and each typically performs better than ESG, and the best WalkSAT variants e.g., Novelty [2]. We have also conducted experiments that show SAPS is similarly effective on MAX-SAT problem instances [5].