Worst-case analysis, 3-SAT decision and lower bounds: Approaches for improved SAT algorithms

Abstract

New methods for worst-case analysis and (3-)SAT decision are presented. The focus lies on the central ideas leading to the improved bound 1:5045 n for 3-SAT decision ((Ku96]; n is the number of variables). The implications for SAT decision in general are discussed and elucidated by a number of hypothesis'. In addition an exponential lower bound for a general class of SAT-algorithms is given and the only possibilities to remain under this bound are pointed out. In this article the central ideas leading to the improved worst-case upper bound 1:5045 n for 3-SAT decision ((Ku96]) are presented. 1) In nine sections the following subjects are treated: 1. \Gauging of branchings": The \-function" and the concept of a \distance function" is introduced, our main tools for the analysis of SAT algorithms, and, as we propose, also a basis for (complete) practical algorithms. 2. \Estimating the size of arbitrary trees": The \-Lemma" is presented, yielding an upper bound for the number of leaves of arbitrary trees, and our central criterion for \good" distance functions is discussed. 3. \The basic idea for the improved 3-SAT algorithm": Starting from the worst-case bounds 1:619 n ((MoSp79], Lu84], MoSp85]) and 1:579 n ((Sch92]), we explain that observation which has been the origin of our investigations. 4. \The reened distance function n ? ": The main tool for the (reened) analysis, the improved distance function, is discussed ((n is the loss of variables, while counts the variation in the number of 2-clauses by means 1) For the history of the subject, see Ku96], and also KuLu96]; for the whole eld of SAT algorithms see GPFW96].