Abstract
We consider a fundamental problem in the theory of branching heuristics for tree-based solvers, applicable e.g. to SAT, #SAT, CSP, #CSP. Such tree-based solvers are used as the cubing-part in the Cube-and-Conquer paradigm, and are thus of renewed interest for general (#)SAT solving. These solvers build at least implicitly a branching (backtracking) tree, with the goal to minimise tree-size. The heuristics are based on evaluating the progress made in a transition from an instance F to some “simplified” \(F'\) by a distance \(d(F,F')\) (the bigger the more progress). When a branching \((F'_1, \dots , F'_k)\) is to be chosen for F, for each possibility we consider its branching tuple t given by \(t_i = d(F, F'_i)\), project it to a single number \(\pi (t)\), and choose a branching with minimal \(\pi (t)\). This paper investigates the choices for \(\pi (t)\), in a theoretical framework. The general theory is reviewed, together with the theoretical result on the “canonical projection” \(\pi (t) = \tau (t)\). Focusing then on binary branchings (\(k=2\), \(t = (a,b)\)), we analyse the asymptotics of \(\tau (a,b)\), and reflect on the whole possible range of binary projections, arriving at first practical possibilities for dynamic heuristics.
O. Kullmann and O. Zaikin—Supported by EPSRC grant EP/S015523/1.
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Kullmann, O., Zaikin, O. (2021). Projection Heuristics for Binary Branchings Between Sum and Product. In: Li, CM., Manyà, F. (eds) Theory and Applications of Satisfiability Testing – SAT 2021. SAT 2021. Lecture Notes in Computer Science(), vol 12831. Springer, Cham. https://doi.org/10.1007/978-3-030-80223-3_21
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