Additive versus Multiplicative Clause Weighting for SAT

This paper examines the relative performance of additive and multiplicative clause weighting schemes for propositional satisfiability testing. Starting with one of the most recently developed multiplicative algorithms (SAPS), an experimental study was constructed to isolate the effects of multiplicative in comparison to additive weighting, while controlling other key features of the two approaches, namely the use of random versus flat moves, deterministic versus probabilistic weight smoothing and multiple versus single inclusion of literals in the local search neighborhood. As a result of this investigation we developed a pure additive weighting scheme (PAWS) which can outperform multiplicative weighting on a range of difficult problems, while requiring considerably less effort in terms of parameter tuning. We conclude that additive weighting shows better scaling properties because it makes less distinction between costs and so considers a larger domain of possible moves. Introduction and Background Clause weighting algorithms for satisfiability testing have formed an important research area since their first introduction in the early 1990s. Since then various improvements have been proposed, resulting in the two best known algorithms of today: the discrete Lagrangian method (DLM) (Wu & Wah 2000) and scaling and probabilistic smoothing (SAPS) (Hutter, Tompkins, & Hoos 2002). While these methods differ in important aspects, both use the same underlying trap avoiding strategy: increasing weights on unsatisfied clauses in local minima and then periodically adjusting weights to maintain reasonable weight differentials during the search. The earliest clause weighting algorithms, such as Breakout (Morris 1993), repeatedly increased weights on unsatisfied clauses and so allowed unrestricted weight growth during the search. Flips were then chosen on the basis of minimizing the combined weight of the unsatisfied clauses. In 1997, Frank proposed a new weight decay algorithm that updated weights on unsatisfied clauses using a combination of a multiplicative decay rate and an additive weight increase. While Frank’s work laid the ground for future advances, his decay scheme produced relatively small improvements over Copyright c 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. earlier weighting approaches. At this point, clause weighting algorithms proved competitive on many smaller problems but were unable to match the performance of faster and simpler heuristics, such as Novelty, on larger problem instances (McAllester, Selman, & Kautz 1997). As a key reason for developing incomplete local search techniques is to solve problems beyond the reach of complete SAT solvers, the poor scalability of clause weighting was a major disadvantage. It was not until the development of DLM that a significant performance gain was achieved. In its simplest form, DLM follows Breakout’s weight increment scheme, but additionally decrements clause weights after a fixed number of increases. DLM also alters the point at which weight is increased by allowing flat moves that leave the weighted cost of the solution unchanged. These flat moves are in turn controlled by a tabu list and by a parameter which limits the total number of consecutive flat moves (Wu & Wah 2000). In empirical tests DLM proved successful at solving a range of random and structured SAT problems, and in particular was able to outperform the best non-weighting algorithms on many larger and more difficult problem instances. In another line of research, Schuurmans and Southey (2000) developed a fully multiplicative weighting algorithm: smoothed descent and flood (SDF). SDF introduced a new method for breaking ties between equal cost flips by additionally considering the number of true literals in satisfied clauses. In situations where no improving moves are available, SDF multiplicatively increases weights on unsatisfied clauses and then normalizes (or smooths) clause weights so that the greatest cost difference between any two flips remains constant. SDF’s reported flip performance was promising in comparison to DLM, but these results did not look at the more difficult problems for which DLM was especially suited. In addition, SDF’s time performance did not compare well, due to the overhead of adjusting weights on all clauses at each local minimum. In subsequent work, SDF evolved into the exponentiated subgradient algorithm (ESG) (Schuurmans, Southey, & Holte 2001), which in turn formed the basis of the scaling and probabilistic smoothing (SAPS) algorithm (Hutter, Tompkins, & Hoos 2002). ESG and SAPS dispensed with SDF’s augmented cost function, and SAPS further improved on the run-time performance of ESG by only smoothing weights periodically, and only increasing weights on violated clauses in a local minimum1. The feature of greatest interest to the current study is that, ignoring the issue of additive versus multiplicative clause weighting, the weight update scheme of SAPS is almost identical in structure to the weight update scheme of DLM: both increase weight when a local minimum is identified (although using different identification criteria), and both periodically adjust weights according to a parameter value that varies for different problems2. SAPS differs from DLM only in using the parameter to probabilistically determine when weight is reduced, whereas DLM deterministically reduces weight after a fixed number of increases. The aim of this study is to investigate whether an additive or multiplicative weight update scheme is better for satisfiability testing. Given that SAPS and DLM both have some claim to be considered as the state-of-the-art in local search for SAT and that both have separately hit upon a similar underlying weighting structure, it now becomes possible to compare additive and multiplicative clause weighting without their relative performance being disguised by differing implementation details. To perform this comparison, we started with the authors’ original version of SAPS and changed it in small steps until it became an effective additive clause weighting algorithm. By examining and empirically testing the effect of each step, we set out to isolate exactly those features that are crucial for the success of each approach. This resulted in the development of a new pure additive weighting scheme (PAWS). As the published results for SAPS have only looked at relatively small problems, we also decided to evaluate SAPS and PAWS on an extended test set that includes a selection of the more difficult problems for which DLM was developed. In the remainder of the paper we describe in more detail the development of PAWS from SAPS and DLM, and then present the results and conclusions of our empirical study. Clause Weighting Algorithms for SAT DLM has been described as “ad hoc” (Schuurmans, Southey, & Holte 2001) and criticized for requiring a large number of parameters to obtain optimum performance. However, DLM has evolved through several versions, the last of which was developed specifically to solve the larger towers of Hanoi and parity learning problems from the DIMACS benchmarks (Wu & Wah 2000). As already discussed, the basic structure of DLM is similar to SAPS, except for the heuristic used to control the taking of flat moves. In addition, although the latest version of DLM has 27 parameters, in practice only three of these require adjustment in the SAT domain. ESG’s approach is to scale and smooth the weight on all clauses in every local minima. Additionally, a third clause weighting algorithm, GLSSAT (Mills & Tsang 1999), uses a similar weight update scheme, additively increasing weights on the least weighted unsatisfied clauses and multiplicatively reducing weights whenever the weight on any one clause exceeds a predefined threshold. Although GLSSAT performed well in comparison with Walksat, it could not match DLM on larger problems and so is not considered further in this study. Of particular interest is that DLM uses a single parameter to control the weighting process (corresponding to in Figure 2), which determines when weights are to be reduced. In contrast, SAPS requires two further parameters ( and ) to determine the amount that weights are multiplicatively scaled or smoothed (in DLM clause weight increases and decreases are implemented by adding or subtracting one). The other two DLM parameters ( and ) are used to control the flat move heuristic: Using the terms from Figure 1, if , DLM will randomly select and flip any "! . Otherwise, if # %$& , and the number of immediately preceding consecutive flat moves is ' and !)(+* $-, , then DLM will randomly select and flip any . /!)( , where !0( contains all flat move literals that have not been flipped in the last 1 moves. Otherwise clause weights are additively updated, as per Figure 2. Although SAPS implements a fairly “pure” weighting algorithm, there are a few implementation details that distinguish it from DLM (see Figure 1). The first is the 203 parameter which probabilistically controls whether a random flip is taken when no improving cost move is available. This acts as an alternative to DLM’s flat move heuristic. The second is that the set of local neighborhood of moves for SAPS contains a single copy of each literal that can make a false clause (i.e. turn it from false to true). In DLM, the neighborhood consists of all literals in all false clauses. This means that if a literal appears in more than one false clause, it will appear more than once in the local neighborhood, thereby increasing the probability that it will be selected. Finally, as noted earlier, SAPS uses probabilistic smoothing when adjusting clause weights, i.e. if 46587:9;9 (=< is set to 5% then there is a 1 in 20 chance that weight will be adjusted after an increase. In contrast, DLM’s third parameter fixes the exact number of increases before weight is decreased, and so represents a deterministic weight reduction scheme. Overall, there is little difference between DLM and SAPS in terms of parameter tuning. While SAPS has four parameters ( ?>@ , 203 and 4A587:9;9 (=< ) and a basic version of DLM has three, in practice at least one of the SAPS parameters can be treated as a constant and the others adjusted to suit (in this study 203 is set at 1%). For both algorithms the process of parameter tuning is time consuming, as optimal performance is highly dependent on the correct settings. This compares poorly with simpler non-weighting algorithms, such as adaptive Walksat (Hoos 2002), which only requires the automatic tuning of a single noise parameter. To address this, a version of SAPS called Reactive SAPS (RSAPS) was developed (Hutter, Tompkins, & Hoos 2002) that automatically adjusts the 4A587:9B9 (=< parameter during the search. However we found this algorithm did not perform as well as a properly tuned SAPS on our problem set, so we did not consider it further. Hence, the main design criticism that can be levelled at DLM is that it relies on a somewhat complex flat move heuristic, whereas SAPS can search purely on the basis of weight guidance (while taking the occasional random move). From this it could be argued that multiplicative weighting is superior to additive weighting because it makes finer distinctions between moves and so avoids the need to procedure SAPS begin generate random starting point for each clause CED do: set clause weight F0DHGJI while solution not found and not timed out do