A Comparative Analysis of Backbone Algorithms for Configurable Software Systems
The backbone of a Boolean formula is the set of literals that must be true in every assignment that satisfies the formula. This concept is fundamental to key operations on variability models, including propagating user configuration decisions to identify implied feature selections, detecting dead features and dead code blocks, and preprocessing formulas to accelerate knowledge compilation into tractable representations such as binary decision diagrams. Despite its importance, previous empirical studies have evaluated backbone algorithms solely on SAT competition formulas (typically engineered to test the limits of SAT solvers), leading to inconsistent conclusions about their performance. This study provides the first comprehensive evaluation of formulas derived from real-world variability models, analyzing 21 configurations of 5 state-of-the-art algorithms on 2,371 formulas from configurable systems ranging from 100 variables and 179 clauses to 186,059 variables and 527,240 clauses. The results indicate that variability model formulas are structurally distinct, with higher clause density but greater clause simplicity. Our research provides clear algorithm selection guidelines: Algorithm 2/3 (iterative with solution filtering) is recommended for formulas with 1,000 or fewer variables, while Algorithm 5 (chunked core-based) with adaptive chunk size selection provides the best practical performance for larger formulas. Also, the results show that filtering heuristics have negligible or negative effects on performance for variability models. Finally, the study identifies a research gap: while Algorithm 5 with optimal chunk size can achieve runtime reductions exceeding 50\% compared to Algorithm 2/3 (the one that product line tools implement), the optimal chunk size varies unpredictably across formulas and cannot currently be estimated, opening directions for future research.