On the Complexity of k-SAT

摘要

The k-SAT problem is to determine if a given k-CNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve k-SAT for k⩾3. Here exponential time means 2δn for some δ>0. In this paper, assuming that, for k⩾3, k-SAT requires exponential time complexity, we show that the complexity of k-SAT increases as k increases. More precisely, for k⩾3, define sk=inf{δ: there exists 2δn algorithm for solving k-SAT}. Define ETH (Exponential-Time Hypothesis) for k-SAT as follows: for k⩾3, sk>0. In this paper, we show that sk is increasing infinitely often assuming ETH for k-SAT. Let s∞ be the limit of sk. We will in fact show that sk⩽(1−d/k) s∞ for some constant d>0. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a k-CNF to the satisfiability of a disjunction of 2εnk′-CNFs in fewer variables for some k′⩾k and arbitrarily small ε>0. We also show that such a disjunction can be computed in time 2εn for arbitrarily small ε>0.