On an online random k-SAT model

• Authors: David Kravitz
• Venue: Random Structures & Algorithms
• Year: 2008

Given n Boolean variables x1,…,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 - ${{1}\over {2^{k}}}$)cn + akn clauses for some constant ak. If these clauses are given offline, it is possible to do much better, (1 - ${{1}\over {2^{k}}}$)cn + Ω($\sqrt{c}$)n can be accepted whp. The question of closing the gap between ak and Ω($\sqrt{c}$) for the online version remained open. This article shows that for any k ≥ 1, any online algorithm will accept less than (1 - ${{1}\over {2^{k}}}$)cn + (ln 2)n k-clauses whp, closing the gap between the constant and Ω($\sqrt{c}$). Furthermore we show that this bound is asymptotically tight as k → ∞. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008