k-SAT Instances Research Articles

Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P(i)∼i−β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P(k)∼k−δ, where δ=1+1/β. This property has been observed in most real-world SAT instances. For β=0, our model extends classical random SAT instances. We prove the existence of a SAT–UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β<1/2 when the clause/variable ratio is m/n=1−2β(1−β)2. We also prove that scale-free random k-SAT instances are unsatisfiable with a high probability when the number of clauses exceeds ω(n(1−β)k). The proof of this result suggests that, when β>1−1/k, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes.

Quantum tunneling and quantum walks as algorithmic resources to solve hard K-SAT instances

We present a new quantum heuristic algorithm aimed at finding satisfying assignments for hard K-SAT instances using a continuous time quantum walk that explicitly exploits the properties of quantum tunneling. Our algorithm uses a Hamiltonian H_A(F) which is specifically constructed to solve a K-SAT instance F. The heuristic algorithm aims at iteratively reducing the Hamming distance between an evolving state {|{psi _j}rangle } and a state that represents a satisfying assignment for F. Each iteration consists on the evolution of {|{psi _j}rangle } (where j is the iteration number) under e^{-iH_At}, a measurement that collapses the superposition, a check to see if the post-measurement state satisfies F and in the case it does not, an update to H_A for the next iteration. Operator H_A describes a continuous time quantum walk over a hypercube graph with potential barriers that makes an evolving state to scatter and mostly follow the shortest tunneling paths with the smaller potentials that lead to a state {|{s}rangle } that represents a satisfying assignment for F. The potential barriers in the Hamiltonian H_A are constructed through a process that does not require any previous knowledge on the satisfying assignments for the instance F. Due to the topology of H_A each iteration is expected to reduce the Hamming distance between each post measurement state and a state {|{s}rangle }. If the state {|{s}rangle } is not measured after n iterations (the number n of logical variables in the instance F being solved), the algorithm is restarted. Periodic measurements and quantum tunneling also give the possibility of getting out of local minima. Our numerical simulations show a success rate of 0.66 on measuring {|{s}rangle } on the first run of the algorithm (i.e., without restarting after n iterations) on thousands of 3-SAT instances of 4, 6, and 10 variables with unique satisfying assignments.

On Super Strong ETH

Multiple known algorithmic paradigms (backtracking, local search and the polynomial method) only yield a 2n(1-1/O(k)) time algorithm for k-SAT in the worst case. For this reason, it has been hypothesized that the worst-case k-SAT problem cannot be solved in 2n(1-f(k)/k) time for any unbounded function f. This hypothesis has been called the "Super-Strong ETH", modelled after the ETH and the Strong ETH.
 It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the "critical threshold", where the clause-to-variable ratio is such that randomly chosen instances are satisfiable with probability 1/2. We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. For example, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in 2n(1-c*log(k)/k) time with high probability (over the choice of the formula and the randomness of the algorithm). 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 (1999). 
 The Unique k-SAT problem is the special case where there is at most one satisfying assignment. Improving prior reductions, we show that the Super-Strong ETHs for Unique k-SAT and k-SAT are equivalent. More precisely, we show the time complexities of Unique k-SAT and k-SAT are very tightly correlated: if Unique k-SAT is in 2n(1-f(k)/k) time for an unbounded f, then k-SAT is in 2n(1-f(k)/(2k)) time.

Focused random walk with probability distribution for SAT with long clauses

Focused random walk (FRW) is one of the most influential paradigm of stochastic local search (SLS) algorithms for the propositional satisfiability (SAT) problem. Recently, an interesting probability distribution (PD) strategy for variable selection was proposed and has been successfully used to improve SLS algorithms, resulting in state-of-the-art solvers. However, most solvers based on the PD strategy only use polynomial function (PoF) to handle the exponential decay and are still unsatisfactory in dealing with medium and huge k-SAT instances at and near the phase transition. The present paper is focused on handling all k-SAT instances with long clauses. Firstly, an extensive empirical study of one state-of-the-art FRW solver WalkSATlm on a wide range of SAT problems is presented with the focus given on fitting the distribution of the break value of variable selected in each step, which turns out to be a Boltzmann function. Using theses case studies as a basis, we propose a pseudo normal function (PNF) to fit the distribution of the break value of variable selected, which is actually a variation of the Boltzmann function. In addition, a new tie-breaking flipping (TBF) strategy is proposed to prevent the same variable from being flipped in consecutive steps. The PNF based PD strategy combined with the TBF strategy lead to a new variable selection heuristic named PNF-TBF. The PNF-TBF heuristic along with a variable allocation value (Vav) function are used to significantly improve ProbSAT, a state-of-the-art SLS solver, leads to a new FRW algorithm dubbed PNFSat, which achieves the state-of-the-art performance on a broad range of huge random 7-SAT instance near the phase transition as demonstrated via the extensive experimental studies. Some further improved versions on top of PNFSat are presented respectively, including PNFSat_alt, which achieves the state-of-the-art performance on the medium 7-SAT instances at the phase transition; PNP as well as an integrated version of these three algorithms, named PDSat, which achieves the state-of-the-art performances on all huge and medium random k-SAT instances with long clauses as demonstrated via the comparative studies using different benchmarks.

Results on a Super Strong Exponential Time Hypothesis

All known SAT-solving paradigms (backtracking, local search, and the polynomial method) only yield a 2n(1−1/O(k)) time algorithm for solving k-SAT in the worst case, where the big-O constant is independent of k. For this reason, it has been hypothesized that k-SAT cannot be solved in worst-case 2n(1−f(k)/k) time, for any unbounded ƒ : ℕ → ℕ. This hypothesis has been called the “Super-Strong Exponential Time Hypothesis” (Super Strong ETH), modeled after the ETH and the Strong ETH. We prove two results concerning the Super-Strong ETH:1. It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the “critical threshold”, where the clause-to-variable ratio is 2k ln 2 −Θ(1). We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. In particular, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in 2n(1−Ω( log k)/k) time, with high probability (over the choice of the formula and the randomness of the algorithm). 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).2. The Unique k-SAT problem is the special case where there is at most one satisfying assignment. It is natural to hypothesize that the worst-case (exponential-time) complexity of Unique k-SAT is substantially less than that of k-SAT. Improving prior reductions, we show the time complexities of Unique k-SAT and k-SAT are very tightly related: if Unique k-SAT is in 2n(1−f(k)/k) time for an unbounded f, then k-SAT is in 2n(1−f(k)(1−ɛ)/k) time for every ɛ > 0. Thus, refuting Super Strong ETH in the unique solution case would refute Super Strong ETH in general.

Streamlining variational inference for constraint satisfaction problems**This article is an updated version of: Grover A, Achim T and Ermon S 2018 Streamlining variational inference for constraint satisfaction problems Advances in Neural Information Processing Systems vol 31, ed S Bengio et al (Red Hook, NY: Curran Associates, Inc) pp 10556–66.

Several algorithms for solving constraint satisfaction problems are based on survey propagation, a variational inference scheme used to obtain approximate marginal probability estimates for variable assignments. These marginals correspond to how frequently each variable is set to true among satisfying assignments, and are used to inform branching decisions during search; however, marginal estimates obtained via survey propagation are approximate and can be self-contradictory. We introduce a more general branching strategy based on streamlining constraints, which sidestep hard assignments to variables. We show that streamlined solvers consistently outperform decimation-based solvers on random k-SAT instances for several problem sizes, shrinking the gap between empirical performance and theoretical limits of satisfiability by on average for .

A probabilistic study of generalized solution concepts in satisfiability testing and constraint programming

We study the probabilistic behaviour of solutions of random instances of the Boolean Satisfiability (SAT) and Constraint Satisfaction Problems (CSPs) that generalize the standard notion of a satisfying assignment. Our analysis focuses on a special type of generalized solutions, the (1,0)-super solutions. For random instances of k-SAT, we establish the exact threshold of the phase transition of the solution probability for k≤3, and give upper and lower bounds on the threshold of the phase transition for the case of k≥4. For CSPs, we derive an upper bound on the threshold of having a (1,0)-super solution asymptotically with probability 1, and establish a condition for the expected number of super solutions to grow exponentially.

Improving configuration checking for satisfiable random k-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 ź 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.

Local Search for Hard SAT Formulas: The Strength of the Polynomial Law

Random k-CNF formulas at the anticipated k-SAT phase-transition point are prototypical hard k-SAT instances. We develop a stochastic local search algorithm and study it both theoretically and through a large-scale experimental study. The algorithm comes as a result of a systematic study that contrasts rates at which a certain measure concentration phenomenon occurs. This study yields a new stochastic rule for local search. A strong point of our contribution is the conceptual simplicity of our algorithm. More importantly, the empirical results overwhelmingly indicate that our algorithm outperforms the state-of-the-art. This includes a number of winners and medalist solvers from the recent SAT Competitions.

Scoring Functions Based on Second Level Score for k-SAT with Long Clauses

It is widely acknowledged that stochastic local search (SLS) algorithms can efficiently find models for satisfiable instances of the satisfiability (SAT) problem, especially for random k-SAT instances. However, compared to random 3-SAT instances where SLS algorithms have shown great success, random k-SAT instances with long clauses remain very difficult. Recently, the notion of second level score, denoted as "score_2", was proposed for improving SLS algorithms on long-clause SAT instances, and was first used in the powerful CCASat solver as a tie breaker. In this paper, we propose three new scoring functions based on score_2. Despite their simplicity, these functions are very effective for solving random k-SAT with long clauses. The first function combines score and score_2, and the second one additionally integrates the diversification property "age". These two functions are used in developing a new SLS algorithm called CScoreSAT. Experimental results on large random 5-SAT and 7-SAT instances near phase transition show that CScoreSAT significantly outperforms previous SLS solvers. However, CScoreSAT cannot rival its competitors on random k-SAT instances at phase transition. We improve CScoreSAT for such instances by another scoring function which combines score_2 with age. The resulting algorithm HScoreSAT exhibits state-of-the-art performance on random k-SAT (k>3) instances at phase transition. We also study the computation of score_2, including its implementation and computational complexity.

Asymmetric Continuous-Time Neural Networks without Local Traps for Solving Constraint Satisfaction Problems

There has been a long history of using neural networks for combinatorial optimization and constraint satisfaction problems. Symmetric Hopfield networks and similar approaches use steepest descent dynamics, and they always converge to the closest local minimum of the energy landscape. For finding global minima additional parameter-sensitive techniques are used, such as classical simulated annealing or the so-called chaotic simulated annealing, which induces chaotic dynamics by addition of extra terms to the energy landscape. Here we show that asymmetric continuous-time neural networks can solve constraint satisfaction problems without getting trapped in non-solution attractors. We concentrate on a model solving Boolean satisfiability (k-SAT), which is a quintessential NP-complete problem. There is a one-to-one correspondence between the stable fixed points of the neural network and the k-SAT solutions and we present numerical evidence that limit cycles may also be avoided by appropriately choosing the parameters of the model. This optimal parameter region is fairly independent of the size and hardness of instances, this way parameters can be chosen independently of the properties of problems and no tuning is required during the dynamical process. The model is similar to cellular neural networks already used in CNN computers. On an analog device solving a SAT problem would take a single operation: the connection weights are determined by the k-SAT instance and starting from any initial condition the system searches until finding a solution. In this new approach transient chaotic behavior appears as a natural consequence of optimization hardness and not as an externally induced effect.

Local search for Boolean Satisfiability with configuration checking and subscore

This paper presents and analyzes two new efficient local search strategies for the Boolean Satisfiability (SAT) problem. We start by proposing a local search strategy called configuration checking (CC) for SAT. The CC strategy results in a simple local search algorithm for SAT called Swcc, which shows promising experimental results on random 3-SAT instances, and outperforms TNM, the winner of SAT Competition 2009.However, the CC strategy for SAT is still in a nascent stage, and Swcc cannot yet compete with Sparrow2011, which won SAT Competition 2011 just after Swcc had been designed. The CC strategy seems too strict in that it forbids flipping those variables even with great scores, if they do not satisfy the CC criterion. We improve the CC strategy by adopting an aspiration mechanism, and get a new variable selection heuristic called configuration checking with aspiration (CCA). The CCA heuristic leads to an improved algorithm called Swcca, which exhibits state-of-the-art performance on random 3-SAT instances and crafted ones.The third contribution concerns improving local search algorithms for random k-SAT instances with k>3. Although the SAT community has made great achievements in solving random 3-SAT instances, the progress lags far behind on random k-SAT instances with k>3. This work proposes a new variable property called subscore, which is utilized to break ties in the CCA heuristic when candidate variables for flipping have the same score. The resulting algorithm CCAsubscore is very efficient for solving random k-SAT instances with k>3, and significantly outperforms other state-of-the-art ones. Combining Swcca and CCAsubscore, we obtain a local search SAT solver called CCASat, which was ranked first in the random track of SAT Challenge 2012.Additionally, we perform theoretical analyses on the CC strategy and the subscore property, and show interesting results on these two heuristics. Particularly, our analysis indicates that the CC strategy is more effective for k-SAT with smaller k, while the subscore notion is not suitable for solving random 3-SAT.

Bose–Einstein condensation in satisfiability problems

This paper is concerned with the complex behavior arising in satisfiability problems. We present a new statistical physics-based characterization of the satisfiability problem. Specifically, we design an algorithm that is able to produce graphs starting from a k-SAT instance, in order to analyze them and show whether a Bose–Einstein condensation occurs. We observe that, analogously to complex networks, the networks of k-SAT instances follow Bose statistics and can undergo Bose–Einstein condensation. In particular, k-SAT instances move from a fit-get-rich network to a winner-takes-all network as the ratio of clauses to variables decreases, and the phase transition of k-SAT approximates the critical temperature for the Bose–Einstein condensation. Finally, we employ the fitness-based classification to enhance SAT solvers (e.g., ChainSAT) and obtain the consistently highest performing SAT solver for CNF formulas, and therefore a new class of efficient hardware and software verification tools.

Maxterm Covering for Satisfiability

This paper presents a novel efficient satisfiability (SAT) algorithm based on maxterm covering. The satisfiability of a clause set is determined in terms of the number of relative maxterms of the empty clause with respect to the clause set. If the number of relative maxterms is zero, it is unsatisfiable, otherwise satisfiable. A set of synergic heuristic strategies are presented and elaborated. We conduct a number of experiments on 3-SAT and k-SAT problems at the phase transition region, which have been cited as the hardest group of SAT problems. Our experimental results on public benchmarks attest to the fact that, by incorporating our proposed heuristic strategies, our enhanced algorithm runs several orders of magnitude faster than the extension rule algorithm, and it also runs faster than zChaff and MiniSAT for most of k-SAT (k≥3) instances.

A quantum differentiation of k-SAT instances

We present a quantum adiabatic algorithm to differentiate between k-SAT instances, those with no solutions and those that have many solutions. The time complexity of the algorithm is a function of the energy gap between the subspace of all 0-eigenvectors (ground states) and the first excited states manifold, and scales polynomially with the number of resources. The idea of gaps between subspaces suggests a new tool to analyze time complexity in adiabatic quantum machines.

Analysis of Local Search Landscapes for k-SAT Instances

Stochastic local search is a successful technique in diverse areas of combinatorial optimisation and is predominantly applied to hard problems. When dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase is helpful for an appropriate parameter adjustment of local search-based procedures. In the present paper, we address parameter estimations in the context of landscapes induced by k-SAT instances: at first, we utilise a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood relation. The objective function is given by the number of satisfied clauses. The procedure provides good approximations of the actual number of local maxima, with a deviation typically around 10%. Secondly, we provide a method for obtaining upper bounds for the average number of local maxima in k-SAT instances. The method allows us to obtain the upper bound \(2^{n-O(\sqrt{n/k})}\) for the average number of local maxima, if m is in the region of 2k· n/k.

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 ( ( k − 1 ) n ) , and presents a k-SAT instance that has Θ ( ( k − 1 ) n ) expected runtime bound.

A Note on the Use of Independent Sets for the k-SAT Problem

An independent set of variables is one in which no two variables occur in the same clause in a given k-SAT instance. Recently, independent sets have obtained more attention. Due to a simple observation we prove that a k-SAT instance over n variables with independent set of size i can be solved in time O(φ_{2(k−1)} (n − i)) where φ_{k (n)} denotes an upper bound on the complexity of solving k-SAT over n variables

Can rare SAT formulae be easily recognized? On the efficiency of message-passing algorithms for K-SAT at large clause-to-variable ratios

For large clause-to-variable ratios, typical K-SAT instances drawn from the uniform distribution have no solution. We argue, based on statistical mechanics calculations using the replica and cavity methods, that rare satisfiable instances from the uniform distribution are very similar to typical instances drawn from the so-called planted distribution, where instances are chosen uniformly between the ones that admit a given solution. It then follows, from a recent article by Feige, Mossel and Vilenchik (2006 Complete convergence of message passing algorithms for some satisfiability problems Proc. Random 2006 pp 339–50), that these rare instances can be easily recognized (in O(log N) time and with probability close to 1) by a simple message-passing algorithm.

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 ? of clauses to variables. The transition takes place at approximately ? = 3.5. We show that for ? > 3.78 with high probability (w.h.p.) random regular 3-SAT formulas are unsatisfiable. Specifically, the events $${\user1{\mathcal{E}}}_{n} $$ hold with high probability if Pr $${\left( {{\user1{\mathcal{E}}}_{n} } \right)} \to 1$$ 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 ? < 2.46, a greedy algorithm finds a satisfying assignment with positive probability.