Super Strong ETH is False for Random k-SAT
share
It has been hypothesized that k-SAT is hard to solve for randomly chosen instances near the "critical threshold", where the clause-to-variable ratio is 2^k 2-θ(1). Feige's hypothesis for k-SAT says that for all sufficiently large clause-to-variable ratios, random k-SAT cannot be refuted in polynomial time. It has also been hypothesized that the worst-case k-SAT problem cannot be solved in 2^n(1-ω_k(1)/k) time, as multiple known algorithmic paradigms (backtracking, local search and the polynomial method) only yield an 2^n(1-1/O(k)) time algorithm. This hypothesis has been called the "Super-Strong ETH", modeled after the ETH and the Strong ETH. Our main result is a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT, for any clause-to-variable ratio. Given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in 2^n(1-Ω( k)/k) time, with high probability. It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1998).
READ FULL TEXT