Some Results on Random Unsatisfiable k-Sat Instances and Approximation Algorithms Applied to Random Structures

In recent years, machine learning and deep learning have made remarkable progress and are now widely applied in various fields, including image classification, autonomous driving, natural language processing, and medical diagnosis. However, training these models requires large datasets, which often contain substantial amounts of sensitive personal information, such as medical records and financial details. Without effective privacy protection measures during model training, the risk of sensitive data leakage increases, potentially resulting in severe privacy violations and a loss of trust. As an innovative data representation technique, the Negative Database has proven to be an effective solution in privacy-sensitive domains. Negative databases can be derived from SAT (Boolean Satisfiability Problem) instances, and the hardness of these instances is directly correlated with the level of data protection provided by the Negative Databases. Developing efficient SAT instance generation algorithms to create harder SAT instances can significantly enhance the privacy protection capabilities of the Negative Database. This paper analyzes the hardness conditions of SAT solvers using the Conflict-Driven Clause Learning strategy and proposes a two-stage SAT instance generation algorithm to generate harder SAT instances. These hard instances not only aid in constructing more secure negative databases for enhanced privacy protection but also provide valuable test cases for evaluating and improving SAT solvers.

Modern CDCL SAT solvers routinely solve very large industrial SAT instances in relatively short periods of time. It is clear that these solvers somehow exploit the structure of real-world instances. However, to-date there have been few results that precisely characterise this structure. In this paper, we provide evidence that the community structure of real-world SAT instances is correlated with the running time of CDCL SAT solvers. It has been known for some time that real-world SAT instances, viewed as graphs, have natural communities in them. A community is a sub-graph of the graph of a SAT instance, such that this sub-graph has more internal edges than outgoing to the rest of the graph. The community structure of a graph is often characterised by a quality metric called Q. Intuitively, a graph with high-quality community structure (high Q) is easily separable into smaller communities, while the one with low Q is not. We provide three results based on empirical data which show that community structure of real-world industrial instances is a better predictor of the running time of CDCL solvers than other commonly considered factors such as variables and clauses. First, we show that there is a strong correlation between the Q value and Literal Block Distance metric of quality of conflict clauses used in clause-deletion policies in Glucose-like solvers. Second, using regression analysis, we show that the the number of communities and the Q value of the graph of real-world SAT instances is more predictive of the running time of CDCL solvers than traditional metrics like number of variables or clauses. Finally, we show that randomly-generated SAT instances with 0.05 ≤ Q ≤ 0.13 are dramatically harder to solve for CDCL solvers than otherwise.

Ordering constraints are analogous to instances of the satisfiability problem in conjunctive normalform, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering iff the relative ordering of its variables obeys the constraint. The naturally arising satisfiability problems are NP-complete for many types of constraints. The present paper seems to be one of the first looking at random ordering constraints. Experimental evidence suggests threshold phenomena as in the case of random k -SAT instances. We prove first that random instances of the cyclic ordering and betweenness constraint have a sharp threshold for unsatisfiability. Second, random instances of the cyclic ordering constraint are satisfiable with high probability if the number of clauses is $\le 1 \times\,\, \sharp$variables.

Solving Satisfiability is at the core of a wide range of applications from Knowledge Representation to Logic Programming to Software and Hardware Verification. One of the models of Satisfiability, the Random Satisfiability problem, has received much attention in the literature both, as a useful benchmark for SAT solvers, and as an exciting mathematical object. In this paper we tackle a somewhat nonstandard type of Random Satisfiability, the one where instances are not chosen uniformly from a certain class of instances, but rather from a certain nontrivial distribution. More precisely, we use so-called Configuration Model, in which we start with a distribution of degrees (the number of occurrences) of a variable, sample the degree of each variable and then generate a random instance with the prescribed degrees. It has been proposed previously that by properly selecting the starting distribution (to be, say, power law or lognorm) one can approximate at least some aspect of `industrial' instances of SAT. Here we suggest an algorithm that solves such problems for a wide range of degree distributions and obtain a necessary and a sufficient condition for the satisfiability of such formulas.

Previous research has shown that 3-SAT problems are easy to solve both when the “constrainedness” (the ratio of the number of clauses to the number of variables) is low and when it is high, abruptly transitioning from easy to hard in a very narrow region of constrainedness. Most of these “phase transition” studies were done on SAT instances that follow uniform random distribution. In such a distribution, variables take part in clauses with uniform probability, and clauses are independent (uncorrelated). The assumptions of uniform random distribution are, however, not satisfied when we consider SAT instances that result from real problems. Our project aims for a deeper understanding of the hardness of SAT problems that arise in practice. In particular, we study two key questions: (1) How does the phase transition behavior change with more realistic and natural distributions of SAT problems? and (2) Can we gain an understanding of the phase transition in terms of the network structure of these SAT problems? Our hypothesis is that the network properties help predict and explain how the easy-to-hard problem transition for realistic SAT problems differs from those for uniform random distribution.

Heuristic average-case analysis of the backtrack resolution of random 3-satisfiability instances

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constantc=c(γ) such that the following holds. LetG= (V, E) be a (pseudo)random directed graph onnvertices and with at least a linear number of edges, and letG′ be a subgraph ofGwith (1/2 + γ)|E| edges. ThenG′ contains a directed cycle of length at least (c−o(1))n. Moreover, there is a subgraphG′′ofGwith (1/2 + γ −o(1))|E| edges that does not contain a cycle of length at leastcn. © 2011 Wiley Periodicals, Inc. J Graph Theory 70: 284–296, 2012

The random 3-satisfiability (3-SAT) problem is in the unsatisfiable (UNSAT) phase when the clause density α exceeds a critical value α(s)≈4.267. Rigorously proving the unsatisfiability of a given large 3-SAT instance is, however, extremely difficult. In this paper we apply the mean-field theory of statistical physics to the unsatisfiability problem, and show that a reduction to 3-XORSAT, which permits the construction of a specific type of UNSAT witnesses (Feige-Kim-Ofek witnesses), is possible when the clause density α>19. We then construct Feige-Kim-Ofek witnesses for single 3-SAT instances through a simple random sampling algorithm and a focused local search algorithm. The random sampling algorithm works only when α scales at least linearly with the variable number N, but the focused local search algorithm works for clause density α>cN(b) with b≈0.59 and prefactor c≈8. The exponent b can be further decreased by enlarging the single parameter S of the focused local search algorithm.

A comparative runtime analysis of heuristic algorithms for satisfiability problems

In combinatorial optimization, a popular approach toNP-hard problems is the design of approximation algorithms. These algorithms typically run in polynomial time and are guaranteed to produce a solution which is within a known multiplicative factor of optimal. Unfortunately, the known factor is often known to be large in pathological instances. Conventional wisdom holds that, in practice, approximation algorithms will produce solutions closer to optimal than their proven guarantees. In this paper, we use the rigorous-analysis-of-heuristics framework to investigate this conventional wisdom.We analyze the performance of 3 related approximation algorithms for the uncapacitated facility location problem (from [Jain, Mahdian, Markakis, Saberi, Vazirani, 2003] and [Mahdian, Ye, Zhang, 2002]) when each is applied to an instances created by placing n points uniformly at random in the unit square. We find that, with high probability, these 3 algorithms do not find asymptotically optimal solutions, and, also with high probability, a simple plane partitioning heuristic does find an asymptotically optimal solution.

Many approximation algorithms have been presented in the last decades for ${\cal NP}$-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms—efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435–472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550–559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

The NP-Completeness theory states that exact and efficient algorithms are unlikely to exist for the class of NP-difficult problems. One way to deal with NP hardness is to relax the optimality requirement and look for solutions instead that are close to the optimum. This is the main idea behind the approximation algorithms, which are called heuristic or metaheuristic. The problem of motion estimation is a process with a high degree of computational complexity, it requires sufficient memory space and execution time. It represents the cost of static, dynamic and video image sequence coding. The main task is to minimize the distortion rate and improve visual quality. This makes research in the field of coding, image compression and video focus on finding efficient algorithms to carry out the estimation of movement in a reasonable time. If a list of images of n elements is analyzed, there are feasible solutions. So, an exhaustive search is too slow, even for small values of the solution space. Therefore, from a practical point of view, it is crucial to have efficient and fast heuristic algorithms that avoid thorough search. In this investigation we design and implement heuristic algorithms, based on the frequency domain, which are applied on the coefficients of the discrete transform of the cosine and wavelets. Also, we propose temporal domain algorithms such as block-matching algorithms, which focus your search on the maximum coincidence of the current image with the reference one. The algorithms used during the implementation of this research work were written with the mathematical programming language MATLAB. In addition, we review the basic concepts of image processing, video, compression algorithms and motion estimation frequently used. The evaluation of the algorithms was carried out with a set of images provided by a previous acquisition system. We show the improvement of visual quality, the amount of compressed or reconstructed information and the behavior of the methods in the search for similarities between pixels or images. Finally, we contribute to the dissemination of new lines of scientific research that lead to the expansion and improvement of the study, the generation of new knowledge, since it is a young area within the Education discipline of Nicaragua.

We study the efficiency of polynomial approximation algorithms for solving NP-hard scheduling problems to minimize total tardiness and total weighted tardiness on one machine. The algorithms are based on PSC-algorithms for the considered problems given in [Zgurovsky & Pavlov, Combinatorial Optimization Problems in Planning and Decision Making: Theory and Applications, Springer, 2019]. They were developed by excluding procedures related to exponential enumeration. The approximation algorithms have a polynomial complexity and can solve problems of any practical dimension. We give a method for determining the maximum possible deviation of the approximate solution of the total tardiness problem from the optimum for each individual problem instance. We present experimental data on the solving time and the actual percentage of the deviation of the functional value from the optimum. The deviation was less than 5% after the execution of the approximate algorithm for the weighted tardiness problem and less than 2.4% for the total tardiness problem. This confirms the high efficiency of the approximation algorithms.

Since the emergence of Artificial Intelligence as a field, planning under uncertainty has been viewed as one of its crucial subareas. Accordingly, the current lack of scalability of probabilistic planning techniques is a major reason why the grand vision of AI has not been fulfilled yet. Besides being an obstacle to advances in AI, scalability issues also hamper the applicability of probabilistic planning to real-world problems. A powerful framework for describing probabilistic planning problems is Markov Decision Processes (MDPs). Informally, an MDP specifies the objectives the agent is trying to achieve, the actions the agent can perform, and the states in which it can end up while working towards the objective. Solving an MDP means finding a policy, i.e. an assignment of actions to states, that allows the agent to achieve the objective. Optimal solution methods, those that look for the “best” policy according to some criterion, typically try to analyze all possible states or a large fraction of them. Since state space sizes of realistic scenarios can be astronomical, these algorithms quickly run out of memory. Fortunately, the mathematical structure of some classes of MDPs has allowed for inventing more efficient algorithms. This is the case for Stochastic Shortest Path (SSP) problems, whose mathematical properties gave rise to a family of algorithms called Find-and-Revise. When used in combination with a heuristic, the members of this family can find a near-optimal, or even optimal, policy while avoiding the analysis of many states. Nonetheless, the sheer number of states in real-world problems forces algorithms based on state-level analysis to exhaust their capabilities much too early, calling for a fundamentally different approach. Moreover, several expressive and potentially useful MDP classes are currently not known to have a mathematical structure for creating efficient approximation techniques. This dissertation advances the state of the art in probabilistic planning in three complementary ways. For SSP MDPs, it derives a novel class of approximate algorithms based on generalizing state analysis information across many states. This information is stored in the form of basis functions that are generated automatically and the number of which is much smaller than the number of states. As a result, the proposed algorithms have a very compact memory footprint and arrive at high-quality solutions faster than their state-based counterparts. In a parallel effort, the dissertation also builds up mathematical apparatus for classes of MDPs that previously had no applicable approximation algorithms. The developed theory enables the extension of the powerful Find-and-Revise paradigm to these MDP classes as well, providing the first memory-efficient algorithms for solving them. Last but not least, the dissertation will apply the proposed theoretical techniques to a large-scale real-world problem, urban traffic routing being one of the candidates.

We analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions. We show that, under a strong probabilistic assumption regarding the distribution of the data in an iteration, the decrease in the primal potential function will be $$\Omega (\rho /\sqrt {log(n)} )$$ with high probability, compared to the guaranteedΩ(1). (ρ ⩾2n is a parameter in the potential function andn is the number of variables.) Under the same assumption, we further show that the objective reduction rate of Dikin's affine scaling algorithm is $$(1 - 1/\sqrt {log(n)} )$$ with high probability, compared to no guaranteed convergence rate.