Projection Heuristics for Binary Branchings Between Sum and Product

Oliver Kullmann, Oleg Zaikin

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.