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 which we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that CircuitSAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, Vertex Cover, are SERF--complete for the class SNP of search problems expressible by second order 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 \Omega\Gamma n) . This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even l...

We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. We show that, for each k, the running time of ResolveSat on a k--CNF formula is significantly better than 2 n , even in the worst case. In particular, we show that the algorithm finds a satisfying assignment of a general satisfiable 3--CNF in time O(2 :448n ) with high probability; where the best previous algorithm [13] has running time O(2 :562n ). We obtain a better upper bound of 2 (2 ln 2\Gamma1)n+o(n) = O(2 0:387n ) for 3CNF that have exactly one satisfying assignment (unique k-SAT). For each k, the bounds for general k-CNF are the best currently known for ...

We present and analyze two simple algorithms for finding satisfying assignments of k-CNFs (Boolean formulae in conjunctive normal form with at most k literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable k-CNF formula F in time O(n 2 jF j2 n\Gamman=k ). The second algorithm is deterministic, and its running time approaches 2 n\Gamman=2k for large n and k. The randomized algorithm is the best known algorithm for k ? 3; the deterministic algorithm is the best known deterministic algorithm for k ? 4. We also show an \Omega\Gamma n 1=4 2 p n ) lower bound on the size of depth 3 circuits of AND and OR gates computing the parity function. This bound is tight up to a constant factor. The key idea used in these upper and lower bounds is what we call the Satisfiability Coding Lemma. This basic lemma shows how to encode satisfying solutions of a k-CNF succinctly. 1 Introduction The problem of ...

We prove the worst-case upper bound 1:5045 n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees. Keywords: 3-SAT, worst-case upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3-SAT decision and prove the worst-case upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...

A new algorithm for testing satisfiability of propositional formulas in conjunctive normal form (CNF) is described. It applies reasoning in the form of certain resolution operations, and identification of equivalent literals. Resolution produces growth in the size of the formula, but within a global quadratic bound; most previous methods avoid operations that produce any growth, and generally do not identify equivalent literals. Computational experience indicates that the method does substantially less "guessing" than previously reported algorithms, while keeping a polynomial time bound on the work done between guesses. Experiments indicate that, for larger problems, the time investment in reasoning returns a profit in reduced searching, and the profit increases with increasing problem size. Experimental data compares six branching strategies of the proposed algorithm on a variety of problems, including several Dimacs benchmarks. These branching strategies were shown to perform differently with statistical signi cance. A new scheme based on Johnson's maximum satisfiability approximation algorithm was found to be the best overall. Both satisfiable and unsatifi able random 3-CNF formulas with 50-283 variables and 4.27 ratio of clauses to variables have been tested; this class is generally acknowledged to be relatively "hard" and

We investigate polynomial reductions and efficient branching rules for algorithms deciding propositional tautologies for DNF and coNP--complete subclasses. Upper bounds on the time complexity are given with exponential part 2 ff\Delta(F ) where (F ) is one of the measures n(F ) = #f variables g, `(F ) = #f literal occurrences g and k(F ) = #f clauses g. We start with a discussion of variants of the algorithms from [Monien/Speckenmeyer85] and [Luckhardt84] with the known upper bound 2 0:695\Deltan for 3-DNF and (roughly) (2 \Delta (1 \Gamma 2 \Gammap )) n for p-DNF, p 3, where p is the maximal clause length, giving now an uniform treatment for all p-DNF including the easy decidable case p 2. Recently for 3-DNF the bound has been lowered to 2 0:5892\Deltan ([K2]; see also [Sch2], [K3]). In this article further improvements are achieved by studying two additional characteristic groups of parameters. The first group differentiates according to the maximal numbers (a; b) of occ...

We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3list -coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3-SAT is equivalent to (2, 3)-CSP while the other problems above are special cases of (3, 2)-CSP. We give a fast algorithm for (3, 2)- CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas. 1 Introduction There has recently been growing interest in analysis of superpolynomial-time algorithms, including algorithms for NP-hard problems such as satisfiability or graph coloring. This interest has multiple causes: . Many important applications can be modeled with these problems, and with the increased speed of modern computers, solved effectively; for instance it is now routine to solve hard 500-variable satisfia...

. New methods for worst-case analysis and (3-)SAT decision are presented. The focus lies on the central ideas leading to the improved bound 1:5045 n for 3-SAT decision ([Ku96]; n is the number of variables). The implications for SAT decision in general are discussed and elucidated by a number of hypothesis'. In addition an exponential lower bound for a general class of SAT-algorithms is given and the only possibilities to remain under this bound are pointed out. In this article the central ideas leading to the improved worst-case upper bound 1:5045 n for 3-SAT decision ([Ku96]) are presented. 1) In nine sections the following subjects are treated: 1. "Gauging of branchings": The " -function" and the concept of a "distance function" is introduced, our main tools for the analysis of SAT algorithms, and, as we propose, also a basis for (complete) practical algorithms. 2. "Estimating the size of arbitrary trees": The " -Lemma" is presented, yielding an upper bound for the number of l...

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 $n algorithm for solving k-SAT]. Define ETH (Exponential-Time Hypothesis) for k-SAT as follows: for k 3, s k>0. In this paper, we show that s k is increasing infinitely often assuming ETH for k-SAT. Let s be the limit of s k. We will in fact show that s k (1&d k) s for some constant d>0. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a k-CNF to the satisfiability of a disjunction of 2 =n k$-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.

In 1980 B. Monien and E. Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses can be checked in time of the order 2^{K/3}. Recently O. Kullmann and H. Luckhardt proved the bound 2^{L/9}, where L is the length of the input formula. The algorithms leading to these bounds (like many other SAT algorithms) are based on splitting, i.e., they reduce SAT for a formula F to SAT for several simpler formulas F1 , F2 , ... , Fm . These algorithms simplify each of F1 , F2 , ... , Fm according to some transformation rules such as the elimination of pure literals, the unit propagation rule etc. In this paper we present a new transformation rule and two algorithms using this rule. These algorithms have the bounds 2^{0.30897K} and 2^{0.10537L}, respectively.