Understanding VSIDS Branching Heuristics in Conflict-Driven Clause-Learning SAT Solvers

Jia Hui Liang
Vijay Ganesh
Ed Zulkoski
Atulan zaman
Krzysztof Czarnecki

Liang J.H., Ganesh V., Zulkoski E., Zaman A., Czarnecki K. (2015) Understanding VSIDS Branching Heuristics in Conflict-Driven Clause-Learning SAT Solvers. In: Piterman N. (eds) Hardware and Software: Verification and Testing. HVC 2015. Lecture Notes in Computer Science, vol 9434. Springer, Cham

7 Interpretation of Results

We began our research by posing a series of questions regarding VSIDS, and we now interpret the results obtained in light of these questions.

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(1) What is special about the class of variables that VSIDS chooses to additively bump? Translation: What is special about the variables selected by vsid?

In the bridge variables experiment (Sect. 3), we showed that VSIDS disproportionately favored bridge variables.

Translation: In the bridge variable experiment (Section 3), we found that vsid had a very high preference for bridge variables .

Even though SAT instances have large number of bridge variables on average, the frequency with which VSIDS picks, bumps, and learns bridge variables is much higher.

Translation: Although SAT instances have a large number of bridge variables on average, VSIDS picks, collides, and learns bridge variables much more frequently .

There is no a priori reason to believe that VSIDS would behave like this. This surprising result, plus a previous result that good community structure correlates with faster solving time [38], suggests CDCL solvers exploit community structure.

Translation: This surprising result, coupled with a previous result, that a good community structure is associated with faster resolution time [38] , shows that CDCL solvers have used the community structure.

More precisely, they target variables linking distinct communities, possibly as a way to solve by divide-and-conquer approach.

Translation: More precisely, their goal is to connect the variables of different communities , which may be a method of solving problems using the divide and conquer method.

In the VSIDS vs. TGC experiments (Sect. 4), we used the Spearman’s rank correlation coefficient to show that the VSIDS and TGC rankings are strongly correlated.

Translation: In the experiment of VSIDS and TGC (Section 4), we use Spearman's rank correlation coefficient to show that VSIDS and TGC rank have a strong correlation .

From our experiments, we can say that for all the VSIDS variants considered in this paper, additive bumping matches with the increase in centrality of the chosen variables.

Translation: From our experiments, we can see that for all VSIDS variables considered in this paper, the additive bumps match the increase in the centrality of the selected variables.

We also observe from our results that the variables that solvers pick for branching have very high TGC rank. The concept of centrality allows us to define in a mathematically precise the intuition many solver developers have had, i.e., that branching on “highly constrained variables” is an effective strategy.

Translation: We also observe from the results that the variables selected by the solver for the branch have a very high TGC rank . The concept of centrality allows us to define the intuition of many solver developers in an accurate mathematical way, that is, branching on "highly constrained variables" is an effective strategy .

Our bridge variable experiment combined with the TGC experiment suggests that VSIDS focuses on high-centrality bridge variables.

Translation: Our bridge variable experiments combined with TGC experiments show that vsid focuses on highly central bridge variables.

(2) What role does multiplicative decay play in making VSIDS so effective? Translation: What role does multiplication decay play in making vsid so effective ?

(Answered by Contribution IV, that in turn led to a new adaptive VSIDS presented as Contribution V.) We show that multiplicative decay is essentially a form of exponential smoothing (Sect. 5).

Translation: We prove that multiplicative decay is essentially a form of exponential smoothing (Section 5).

We add an explanation as to why this is important, namely, that exponential smoothing favors variables that persistently occur in conflicts and this is a better strategy for root-cause analysis.

Translation: We have added an explanation why this is important, that is, exponential smoothing is conducive to variables that continue to occur in conflict, which is a better strategy for root cause analysis .

We designed a new VSIDS technique, we call adaptVSIDS, based on the above results, wherein we rapidly decay the VSIDS activity if the learnt clause LBDs are large (Sect. 6). We showed that this technique is better than mVSIDS on the SAT Competition 2013 benchmark.

Translation: According to the above results, if the LBDs of the learned clauses are large, the activity of VSIDS decays rapidly .

(3) Is VSIDS temporally and spatially focused? (Answered by Contribution II.) Translation: Does vsid focus on time and space?

We show that VSIDS exhibits spatial focus and temporal focus (Sect. 3), forms of locality in search. While there has been speculation among solver researchers that CDCL with VSIDS solvers perform local search, we precisely define spatial and temporal locality in terms of the community structure.

Translation: We found that VSIDS exhibits spatial focus and temporal focus (Section 3), the local form of search. Although some researchers speculate that the CDCL and VSIDS solvers perform local search, we accurately define the locality in space and time according to the community structure .

8 Related Work

Marques-Silva and Sakallah are credited with inventing the CDCL technique [34]. The original VSIDS heuristic was invented by the authors of Chaff [36].

Armin Biere [8] described the low-pass filter behavior of VSIDS, and Huang et al. [26] stated that VSIDS is essentially an EMA.

Translation: Armin Biere [8] describes the low-pass filtering behavior of VSIDS , Huang et al. [26] believe that VSIDS is essentially EMA (Exponential Moving Average)

Katsirelos and Simon [30] were the first to publish a connection between eigenvector centrality and branching heuristics. In their paper [30], the authors computed eigenvector centrality (via Google PageRank) only once on the original input clauses and showed that most of the decision variables have higher than average centrality.

Translation: Katsirelos and Simon [30] first published the connection between feature vector centrality and branch heuristics .

Translation: The author calculated the feature vector centrality only once for the original input clause ( via Google PageRank ). The results show that the centrality of most decision variables is higher than the average centrality .

Also, it bears stressing that their definition of centrality is not temporal. By contrast, our results correlate VSIDS ranking with temporal degree and eigenvector centrality, and show the correlation holds dynamically throughout the run of the solver.

Translation: In addition, it needs to be emphasized that their definition of centrality is not temporary.

Translation: Our results link the vsid ranking to the degree of time and the centrality of the feature vector , and dynamically show the correlation throughout the solution process.

Also, we noticed that the correlation is also significantly stronger after extending centrality with temporality. Simon and Katsirelos do hypothesize that VSIDS may be picking bridge variables (they call them fringe variables). However, they do not provide experimental evidence for this.

Translation: In addition, we have noticed that after extending the centrality and timeliness, the correlation is also significantly enhanced . Simon and Katsirelos hypothesized that vsid might choose bridging variables (they call them marginal variables). However, they did not provide experimental evidence for this.

To the best of our knowledge, we are the first to establish the following results regarding VSIDS: first, VSIDS picks, bumps, and learns high-centrality bridge variables; second, VSIDS-influenced search is more spatially and temporally focused than other branching heuristics we considered; third, explain the importance of EMA (multiplicative decay) to the effectiveness of VSIDS; and fourth, invent a new adaptive VSIDS branching heuristic based on our observations.

Translation: To the best of our knowledge, we first determined the following results regarding vsid:

Translation: First, vsid picking, bumping and learning highly central bridging variables ;

Translation: Secondly, compared to other branch heuristics we consider, searches influenced by vsid pay more attention to space and time ;

Translation: Third, explain the importance of EMA (multiplicative decay) to the effectiveness of vsid (essentially a form of exponential smoothing);

Translation: Based on our observations, invent a new adaptive vsid branch heuristic.

sat solver is a big data system

Understanding VSIDS Branching Heuristics in Conflict-Driven Clause-Learning SAT Solvers

7 Interpretation of Results

8 Related Work

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