On Extending Two Threshold 3 - SAT Algorithms to Non - ThresholdAlgorithms by Attaching the Unit Clause

Many of the proofs of lower-bounds on the conjectured satissability threshold value, c , for 3-SAT are based on probabilistic analyses of \iterative Davis-Putnam style" algorithms (IDPS algorithms) or slight modiications of IDPS algorithms 4], 5], 1], 9]. Let PSAT A (m; n; 3) denote the probability that algorithm A nds a satisfying truth assignment for a randomly generated instance (by the xed clause length model) where m; n are the number of clauses and variables, respectively. The probabilistic analyses proceed by nding a t 1 > 0 such that for all 0 < c < t 1 , lim n!1 PSAT A (dcne; n; 3) = 1. The result is that t 1 is a lower-bound on the conjectured value of c. However, a recent result by Friedgut 8] implies that nding a ^ t 1 > 0 such that for all 0 < c < ^ t 1 , lim n!1 PSAT A (dcne; n; 3) > 0 is suu-cient to establish ^ t 1 as a lower-bound on the conjectured value of c. An IDPS algorithm, A1, is said to exhibit a threshold if there exists a t 1 > 0 such that for all 0 < c < t 1 ; lim n!1 PSAT A1 (dcne; n; 3) = 1 and for all t 1 < c; lim n!1 PSAT A (dcne; n; 3) = 0. If A1 can be \extended" to a non-threshold algorithm, A2, such that: there exists c 2 > t 1 where for all 0 < c < c 1 ; lim n!1 PSAT A2 (dcne; n; 3) > 0, then A2 establishes a greater lower-bound on the conjectured value of c than A1 (see gure 1). The broad goal of this paper is to investigate techniques by which threshold IDPS algorithms can be extended to non-threshold algorithms which establish a greater lower-bound on the conjectured value of c. As a rst step toward this goal we investigate the technique of attaching the unit clause rule to IDPS algorithms, PURE and WEIGHT to produce algorithms , UNIT-PURE and UNIT-WEIGHT, respectively. PURE uses the pure literal rule as its literal selection heuristic. WEIGHT uses a weighted literal heuristic developed by Johnson 12]. From experiments, PURE and WEIGHT seem to exhibit a threshold while UNIT-PURE and UNIT-WEIGHT do not. Further, UNIT-PURE and UNIT-WEIGHT both seem to establish a greater lower-bound on the conjectured value …