On an online random k-SAT model

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 which 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 2k )cn+zkn clauses for some constant zk. If these clauses are given offline, it is possible to do much better, (1− 1 2k )cn+Ω( √ c)n can be accepted whp. The question of closing the gap between zk and Ω( √ c) for the online version was posed by Coppersmith, Gamarnik, Hajiaghayi, and Sorkin. This paper shows that for any k ≥ 1, any online algorithm will accept less than (1 − 1 2k )cn + (ln 2)n k-clauses whp, furthermore we show that this bound is asymptotically tight as k → ∞.