Fundaments of Branching Heuristics

The topic of this chapter is to provide foundations for “branching heuristics”. The whole field of “heuristics” is very diverse, and we will concentrate on a specific part, where developments in the last four decades can be comprised in what actually deserves to be called a “theory”. A full version of this chapter, containing all proofs and extensive examples, is available in [Kul08a]. The notion of a “heuristics” is fundamental for the field of Artificial Intelligence. The (early) history of the notion of “heuristics” is discussed in [BF81], Section II.A, and the usage of this notion in the SAT literature follows their definition of a heuristics as a method using additional information to restrict the search space size, though in this chapter we consider a restricted context, where completeness is not an issue, but the heuristical component of the search process affects only resource usage, not correctness. Furthermore we only study a specific form of search processes here, namely backtracking search. We consider the situation where we have a problem instance F where all direct (“efficient”) methods fail, and so F has to be split into subproblems. In the context of this chapter we basically assume that the method for splitting F is already given, yielding possible “branchings” F ; F1, . . . , Fm, splitting F into m “subproblems” Fi, and the task of the heuristic is to compare different branchings and to find the “best” branching among them. Let us assume that we have given three branchings to compare: