A deterministic (2−2/(k+1))n algorithm for k-SAT based on local search

Local search is widely used for solving the propositional satisfiability problem. Papadimitriou (1991) showed that randomized local search solves 2-SAT in polynomial time. Recently, Schöning (1999) proved that a close algorithm for k-SAT takes time (2−2/k)ⁿ up to a polynomial factor. This is the best known worst-case upper bound for randomized 3-SAT algorithms (cf. also recent preprint by Schuler et al.).We describe a deterministic local search algorithm for k-SAT running in time (2−2/(k+1))ⁿ up to a polynomial factor. The key point of our algorithm is the use of covering codes instead of random choice of initial assignments. Compared to other “weakly exponential” algorithms, our algorithm is technically quite simple. We also describe an improved version of local search. For 3-SAT the improved algorithm runs in time 1.481ⁿ up to a polynomial factor. Our bounds are better than all previous bounds for deterministic k-SAT algorithms.     

A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart

A simple probabilistic algorithm for solving the NP-complete problem k -SAT is reconsidered. This algorithm follows a well-known local-search paradigm: randomly guess an initial assignment and then, guided by those clauses that are not satisfied, by successively choosing a random literal from such a clause and changing the corresponding truth value, try to find a satisfying assignment. Papadimitriou [11] introduced this random approach and applied it to the case of 2-SAT, obtaining an expected O(n ² ) time bound. The novelty here is to restart the algorithm after 3n unsuccessful steps of local search. The analysis shows that for any satisfiable k -CNF formula with n variables the expected number of repetitions until a satisfying assignment is found this way is (2? (k-1)/ k) ⁿ . Thus, for 3-SAT the algorithm presented here has a complexity which is within a polynomial factor of (\frac 4 3 ) ⁿ . This is the fastest and also the simplest among those algorithms known up to date for 3-SAT achieving an o(2 ⁿ ) time bound. Also, the analysis is quite simple compared with other such algorithms considered before.     

Randomized Algorithms for 3-SAT

Schoning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n)·(4/3)ⁿ = O(1.334ⁿ). In this paper we present randomized algorithms and show that one of them has O(1.3302ⁿ) expected running time, improving Schoning's algorithm. (Note. At this point, the fastest randomized algorithm for 3-SAT is the one given by Iwama and Tamaki that runs in O(1.324ⁿ).)     

A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart

Abstract. A simple probabilistic algorithm for solving the NP-complete problem k -SAT is reconsidered. This algorithm follows a well-known local-search paradigm: randomly guess an initial assignment and then, guided by those clauses that are not satisfied, by successively choosing a random literal from such a clause and changing the corresponding truth value, try to find a satisfying assignment. Papadimitriou [11] introduced this random approach and applied it to the case of 2-SAT, obtaining an expected O(n ² ) time bound. The novelty here is to restart the algorithm after 3n unsuccessful steps of local search. The analysis shows that for any satisfiable k -CNF formula with n variables the expected number of repetitions until a satisfying assignment is found this way is (2⋅ (k-1)/ k) ⁿ . Thus, for 3-SAT the algorithm presented here has a complexity which is within a polynomial factor of (\frac 4 3 ) ⁿ . This is the fastest and also the simplest among those algorithms known up to date for 3-SAT achieving an o(2 ⁿ ) time bound. Also, the analysis is quite simple compared with other such algorithms considered before.     

Improving configuration checking for satisfiable random <Emphasis Type="Italic">k</Emphasis>-SAT instances

Local search algorithms based on the Configuration Checking (CC) strategy have been shown to be efficient in solving satisfiable random k-SAT instances. The purpose of the CC strategy is to avoid the cycling problem, which corresponds to revisiting already flipped variables too soon. It is done by considering the neighborhood of the formula variables. In this paper, we propose to improve the CC strategy on the basis of an empirical study of a powerful algorithm using this strategy. The first improvement introduces a new and simple criterion, which refines the selection of the variables to flip for the 3-SAT instances. The second improvement is achieved by using the powerful local search algorithm Novelty with the adaptive noise setting. This algorithm enhances the efficiency of the intensification and diversification phases when solving k-SAT instances with k ≥ 4. We name the resulting local search algorithm Ncca+ and show its effectiveness when treating satisfiable random k-SAT instances issued from the SAT Challenge 2012. Ncca+ won the bronze medal of the random SAT track of the SAT Competition 2013.     

Exact Algorithms for MAX-SAT

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given a boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2ⁿ “steps”? Here, n is the number of distinct variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that a popular algorithm based on branch-and-bound is bounded by O(2ⁿ) in time, where n is the maximum number of occurrences of any variable in the input.When the input formula is in 2-CNF, that is, each clause has at most two literals, MAX-SAT becomes MAX-2-SAT and the decision version of MAX-2-SAT is still NP-complete. The best bound of the known algorithms for MAX-2-SAT is O(m2^m/5), where m is the number of clauses. We propose an efficient decision algorithm for MAX-2-SAT whose time complexity is bound by O(n2ⁿ). This result is substantially better than the previously known results. Experimental results also show that our algorithm outperforms any algorithm we know on MAX-2-SAT.     

On a simple randomized algorithm for finding a 2-factor in sparse graphs

We analyze the performance of a simple randomized algorithm for finding 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of degree at most three. For the directed case, the algorithm finds a 2-factor in O(n²) expected time. The analysis of our algorithm is based on random walks on the line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [On selecting a satisfying truth assignment, in: Proc. 32nd Annual IEEE Symp. on the Foundations of Computer Science (FOCS), 1991, p. 163]. For the undirected case, the algorithm finds a 2-factor in O(n³) expected time. We also analyze random versions of these graphs and show that cycles of length Ω(n/logn) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [Finding hidden Hamilton cycles, Random Structures Algorithms 5 (1994) 395] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [On approximating the longest path in a graph, Algorithmica 18 (1997) 82].     

A comparative runtime analysis of heuristic algorithms for satisfiability problems

The satisfiability problem is a basic core NP-complete problem. In recent years, a lot of heuristic algorithms have been developed to solve this problem, and many experiments have evaluated and compared the performance of different heuristic algorithms. However, rigorous theoretical analysis and comparison are rare. This paper analyzes and compares the expected runtime of three basic heuristic algorithms: RandomWalk, (1+1) EA, and hybrid algorithm. The runtime analysis of these heuristic algorithms on two 2-SAT instances shows that the expected runtime of these heuristic algorithms can be exponential time or polynomial time. Furthermore, these heuristic algorithms have their own advantages and disadvantages in solving different SAT instances. It also demonstrates that the expected runtime upper bound of RandomWalk on arbitrary k-SAT (k?3) is O(n(k−1)), and presents a k-SAT instance that has Θ(n(k−1)) expected runtime bound.     

Almost 2-SAT is fixed-parameter tractable

We consider the following problem. Given a 2-cnf formula, is it possible to remove at most k clauses so that the resulting 2-cnf formula is satisfiable? This problem is known to different research communities in theoretical computer science under the names Almost 2-SAT, All-but-k 2-SAT, 2-cnf deletion, and 2-SAT deletion. The status of the fixed-parameter tractability of this problem is a long-standing open question in the area of parameterized complexity. We resolve this open question by proposing an algorithm that solves this problem in O(k¹⁵×k×m³) time showing that this problem is fixed-parameter tractable.     

More efficient two-mode stochastic local search for random 3-satisfiability

Stochastic local search (SLS) is a popular paradigm in incomplete solving for the Boolean satisfiability problem (SAT). Most SLS solvers for SAT switch between two modes, i.e., the greedy (intensification) mode and the diversification mode. However, the performance of these two-mode SLS algorithms lags far behind on solving random 3-satisfiability (3-SAT) problem, which is a significant special case of the SAT problem. In this paper, we propose a new hybrid scoring function called M C based on a linear combination of a greedy property m a k e and a diversification property C o n f T i m e s, and then utilize M C to develop a new two-mode SLS solver called CCMC. To evaluate the performance of CCMC, we conduct extensive experiments to compare CCMC with five state-of-the-art two-mode SLS solvers (i.e., Sparrow2011, Sattime2011, EagleUP, gNovelty+PCL and CCASat) on a broad range of random 3-SAT instances, including all large 3-SAT ones from SAT Competition 2009 and SAT Competition 2011 as well as 200 generated satisfiable huge random 3-SAT ones. The experiments illustrate that CCMC obviously outperforms its competitors, indicating the effectiveness of CCMC. We also analyze the effectiveness of the underlying ideas in CCMC and further improve the performance of CCMC on solving random 5-SAT instances.     

MAX-k-SAT的PTAS归约等价性

通过构造适当的极小不可满足公式,利用子句拼接技术,引入了一个一般化的从k-CNF公式(k≥3)到3-CNF公式之间的归约转换。基于该转换,给出了一个真值指派的转换算法,并证明了MAX-k-SAT与MAX-3-SAT是PTAS归约等价的。因此,对于k,t≥3,MAX-k-SAT与MAX-t-SAT是PTAS归约等价的。     

Optimal 2-constraint satisfaction via sum-product algorithms

We show that for a given set of m pairwise constraints over n variables, a variable assignment that satisfies maximally many m constraints (MAX-2-CSP) can be found in time, where d is the maximum number of states per variable, and ω<2.376 is the matrix product exponent over a ring; the notation O^∗ suppresses factors polylogarithmic in m and n. As a corollary, MAX-2-SAT can be solved in O^∗(nmn^1.732) time. This improves on a recent result by Williams [R. Williams, A new algorithm for optimal 2-constraint satisfaction and its implications, Theoret. Comput. Sci. 348 (2-3) (2005) 357-365] by reducing the polynomial factor from nm³ to about nm.     

Regular Random k-SAT: Properties of Balanced Formulas

We consider a model for generating random k-SAT formulas, in which each literal occurs approximately the same number of times in the formula clauses (regular random and k-SAT). Our experimental results show that such regular random k-SAT instances are much harder than the usual uniform random k-SAT problems. This is in agreement with other results that show that more balanced instances of random combinatorial problems are often much more difficult to solve than uniformly random instances, even at phase transition boundaries. There are almost no formal results known for such problem distributions. The balancing constraints add a dependency between variables that complicates a standard analysis. Regular random 3-SAT exhibits a phase transition as a function of the ratio α of clauses to variables. The transition takes place at approximately α = 3.5. We show that for α > 3.78 with high probability (w.h.p.) random regular 3-SAT formulas are unsatisfiable. Specifically, the events hold with high probability if Pr when n → ∞. We also show that the analysis of a greedy algorithm proposed by Kaporis et al. for the uniform 3-SAT model can be adapted for regular random 3-SAT. In particular, we show that for formulas with ratio α < 2.46, a greedy algorithm finds a satisfying assignment with positive probability.     

Molecular Computing, Bounded Nondeterminism, and Efficient Recursion

The maximum number of strands used is an important measure of a molecular algorithm's complexity. This measure is also called the volume used by the algorithm. Every problem that can be solved by an NP Turing machine with b(n) binary nondeterministic choices can be solved by molecular computation in a polynomial number of steps, with four test tubes, in volume 2 ^b(n) . We identify a large class of recursive algorithms that can be implemented using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3-SAT, independent set, and 3-colorability. Received May 5, 1997; revised March 24, 1998.     

Complexity of Approximating CSP with Balance / Hard Constraints

We study two natural extensions of Constraint Satisfaction Problems (CSPs). Balance-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of Hard-Max-CSP consists of soft constraints and hard constraints, and the goal is to maximize the weight of satisfied soft constraints while satisfying all the hard constraints. These two extensions contain many fundamental problems not captured by CSPs, and challenge traditional theories about CSPs in a more general framework. Max-2-SAT and Max-Horn-SAT are the only two nontrivial classes of Boolean CSPs that admit a robust satisfibiality algorithm, i.e., an algorithm that finds an assignment satisfying at least (1 ? g(ε)) fraction of constraints given a (1 ? ε)-satisfiable instance, where g(ε) → 0 as ε → 0, and g(0) = 0. We prove the inapproximability of these problems with balance or hard constraints, showing that each variant changes the nature of the problems significantly (in different ways). For instance, deciding whether an instance of 2-SAT admits a balanced assignment is NP-hard, and for Max-2-SAT with hard constraints, it is hard to find a constant-factor approximation even on (1 ? ε)-satisfiable instances (in particular, the version with hard constraints does not admit a robust satisfiability algorithm). We also study hardness results for a certain CSP over a larger domain capturing ordering constraints: we show that hard constraints rule out constant-factor approximation algorithms. All our hardness results are almost optimal — they completely rule out algorithms with certain properties, or can be matched by simple extensions to existing algorithms.     

Masking traveling beams: Optical solutions for NP-complete problems, trading space for time

Two architectures for optical processors designed to solve instances of NP-Complete problems, trading space for time, are suggested. The first approach mimics the traveling salesman by an exponential number of traveling beams, that simultaneously examine the different possible paths. The other approach uses a pre-processing stage in which O(n²) masks consisting of an exponential number of locations, are constructed; each representing a different edge in the graph. The choice and combination of the appropriate (small) subset of these masks yields the solution. The solution is rejected in cases where the combination of these masks completely blocks the light and accepted otherwise. We present detailed designs for basic primitives of the optical processor. We propose designs for solving instances of Hamiltonian path, Traveling Salesman, Clique, Independent Set, Vertex Cover, Partition, 3-SAT, 3D-matching, and the Permanent.     

一类可分离SAT问题的O(1.890n)精确算法

布尔可满足性问题（SAT）是指对于给定的布尔公式,是否存在一个可满足的真值指派.这是第1个被证明的NP完全问题,一般认为不存在多项式时间算法,除非P=NP.学者们大都研究了子句长度不超过k的SAT问题（k-SAT）,从全局搜索到局部搜索,给出了大量的相对有效算法,包括随机算法和确定算法.目前,最好算法的时间复杂度不超过O（（2-2/k）ⁿ）,当k=3时,最好算法时间复杂度为O（1.308ⁿ）.而对于更一般的与子句长度k无关的SAT问题,很少有文献涉及.引入了一类可分离SAT问题,即3-正则可分离可满足性问题（3-RSSAT）,证明了3-RSSAT是NP完全问题,给出了一般SAT问题3-正则可分离性的O（1.890ⁿ）判定算法.然后,利用矩阵相乘算法的研究成果,给出了3-RSSAT问题的O（1.890ⁿ）精确算法,该算法与子句长度无关.     

Mapping lake CDOM by satellite remote sensing

Given the importance of coloured dissolved organic matter (CDOM) for the structure and function of lake ecosystems, a method to estimate the amount of CDOM in lake waters over large geographic areas would be highly desirable. Advanced Land Imager (ALI) images were acquired in southern Finland (in 2002) and southern Sweden (in 2003) together with in situ measurements of bio-optical properties of 34 lakes (39 measuring stations). Based on this dataset, a band-ratio type algorithm was developed using ALI band 2 and band 3 for estimating CDOM content (absorption of filtrated water at 420 nm) in lakes. Correlation between in situ measured CDOM and the remote sensing estimate of CDOM was high, r²=0.73. The CDOM retrieval algorithm obtained on the basis of two images and in situ data was validated on a third ALI image (eastern Finland, 2002) that was available in the ALI image archive. In situ water-colour monitoring data from 22 lakes (27 measuring stations) in the third image were available in a database of the Finnish Environment Administration. The water-colour data were converted to CDOM absorption values, which were then compared to the results from a third ALI image. The correlation between remotely estimated and in situ CDOM values in the algorithm validation image was high, r²=0.83. These results support the conclusion that CDOM content in lakes over a wide range of concentrations (a_CDOM(420) between 0.68 and 11.13 m⁻¹) can be mapped using Advanced Land Imager data.     

Computing sparse permanents faster

Bax and Franklin (2002) gave a randomized algorithm for exactly computing the permanent of any n×n zero-one matrix in expected time exp[−Ω(n1/3/(2lnn))]n². Building on their work, we show that for any constant C>0 there is a constant ?>0 such that the permanent of any n×n (real or complex) matrix with at most Cn nonzero entries can be computed in deterministic time n(2−?) and space O(n). This improves on the Ω(n²) runtime of Ryser's algorithm for computing the permanent of an arbitrary real or complex matrix.