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 2 $n for some $>0. In this paper, assuming that, for k 3, k-SAT requires exponential time complexity, we show that the complexity of k-SAT increases as k increases. More precisely, for k 3, define s k =inf[$: there exists 2 $n algorithm for solving k-SAT]. Define ETH (Exponential-Time Hypothesis) for k-SAT as follows: for k 3, s k >0. In this paper, we show that s k is increasing infinitely often assuming ETH for k-SAT. Let s be the limit of s k. We will in fact show that s k (1&dÂk) s for some constant d>0. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a k-CNF to the satisfiability of a disjunction of 2 =n k$-CNFs in fewer variables for some k$ k and arbitrarily small =>0. We also show that such a disjunction can be computed in time 2 =n for arbitrarily small =>0. 2001 Academic Press Although all NP-complete problems are equivalent as far as the existence of polynomial-time algorithm is concerned, there is wide variation in the worst-case complexity of known algorithms for these problems. For example, there have been several algorithms for maximum independent set [6, 12, 17, 18], and the best of these takes time 1.2108 n in the worst-case [12]. Recently, a 3-coloring algorithm with 1.3446 n worst-case time complexity is presented [2] and it is known that k-coloring can be solved in 2.442 n time [8]. However, it is not known what, if any, relationships exist among the worst-case complexities of various problems. In this paper, we examine the complexity of k-SAT, and derive a relationship that governs
For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction that we call Sub-exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that Circuit-SAT is SERF-complete for all NP-search problems, and that for any fixed k \ 3, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, and Vertex Cover, are SERF-complete for the class SNP of search problems expressible by secondorder existential formulas whose first-order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others.We also look at the issue of proving strongly exponential lower bounds for AC 0 , that is, bounds of the form 2 W(n). This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even limited (at most n e ) fan-in for bottom-level gates would imply a nonlinear size lower bound for logarithmic depth circuits. We show that with high probability even random degree 2 GF(2) polynomials require strongly exponential size for S k 3 circuits for k=o(log log n). We thus exhibit a much smaller space of 2functions such that almost every function in this class requires strongly exponential size S k 3 circuits. As a corollary, we derive a pseudorandom generator (requiring O(n 2 ) bits of advice) that maps n bits into a larger number of bits so that computing parity on the range is hard for S k 3 circuits. Our main technical lemma is an algorithm that, for any fixed e > 0, represents an arbitrary k-CNF formula as a disjunction of 2 en k-CNF formulas that are sparse, that is, each disjunct has O(n) clauses.
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