On an online random k-SAT model

Given n Boolean variables x(1),...,xn, a k-clause is a disjunction of k literals, where a literal is a variable or its negation. Suppose random k-clauses are generated one at a time and an online algorithm accepts or rejects each clause as it is generated. Our goal is to accept as many randomly generated k-clauses as possible with the condition that it must be possible to satisfy every clause that is accepted. When cn random k-clauses on n variables are given, a natural online algorithm known as Online-Lazy accepts an expected (1-1/2(k))cn + a(k)n clauses for some constant a(k). If these clauses are given offline, it is possible to do much better, (1-1/2(k))cn + Omega (root c)n can be accepted whp. The question of closing the gap between ak and Omega (root c) for the online version remained open. This article shows that for any k >= 1, any online algorithm will accept less than (1-1/2(k))cn + (In 2)n k-clauses whp, closing the gap between the constant and Omega (root c). Furthermore we show that this bound is asymptotically tight as k -> infinity. (c) 2007 Wiley Periodicals, Inc.