Improve SAT-solving with Machine Learning

In this project, we aimed to improve the runtime of Minisat, a Conflict-Driven Clause Learning (CDCL) solver that solves the Propositional Boolean Satisfiability (SAT) problem. We first used a logistic regression model to predict the satisfiability of propositional boolean formulae after fixing the values of a certain fraction of the variables in each formula. We then applied the logistic model and added a preprocessing period to Minisat to determine the preferable initial value (either true or false) of each boolean variable using a Monte-Carlo approach. Concretely, for each Monte-Carlo trial, we fixed the values of a certain ratio of randomly selected variables, and calculated the confidence that the resulting sub-formula is satisfiable with our logistic regression model. The initial value of each variable was set based on the mean confidence scores of the trials that started from the literals of that variable. We were particularly interested in setting the initial values of the backbone variables correctly, which are variables that have the same value in all solutions of a SAT formula. Our Monte-Carlo method was able to set 78% of the backbones correctly. Excluding the preprocessing time, compared with the default setting of Minisat, the runtime of Minisat for satisfiable formulae decreased by 23%. However, our method did not outperform vanilla Minisat in runtime, as the decrease in the conflicts was outweighed by the long runtime of the preprocessing period.Comment: 2 pages, SIGCSE SRC 201

Phase Transition and Network Structure in Realistic SAT Problems

A fundamental question in Computer Science is understanding when a specific class of problems go from being computationally easy to hard. Because of its generality and applications, the problem of Boolean Satisfiability (aka SAT) is often used as a vehicle for investigating this question. A signal result from these studies is that the hardness of SAT problems exhibits a dramatic easy-to-hard phase transition with respect to the problem constrainedness. Past studies have however focused mostly on SAT instances generated using uniform random distributions, where all constraints are independently generated, and the problem variables are all considered of equal importance. These assumptions are unfortunately not satisfied by most real problems. Our project aims for a deeper understanding of hardness of SAT problems that arise in practice. We study two key questions: (i) How does easy-to-hard transition change with more realistic distributions that capture neighborhood sensitivity and rich-get-richer aspects of real problems and (ii) Can these changes be explained in terms of the network properties (such as node centrality and small-worldness) of the clausal networks of the SAT problems. Our results, based on extensive empirical studies and network analyses, provide important structural and computational insights into realistic SAT problems. Our extensive empirical studies show that SAT instances from realistic distributions do exhibit phase transition, but the transition occurs sooner (at lower values of constrainedness) than the instances from uniform random distribution. We show that this behavior can be explained in terms of their clausal network properties such as eigenvector centrality and small-worldness (measured indirectly in terms of the clustering coefficients and average node distance)

Satisfiability, sequence niches, and molecular codes in cellular signaling

Biological information processing as implemented by regulatory and signaling networks in living cells requires sufficient specificity of molecular interaction to distinguish signals from one another, but much of regulation and signaling involves somewhat fuzzy and promiscuous recognition of molecular sequences and structures, which can leave systems vulnerable to crosstalk. This paper examines a simple computational model of protein-protein interactions which reveals both a sharp onset of crosstalk and a fragmentation of the neutral network of viable solutions as more proteins compete for regions of sequence space, revealing intrinsic limits to reliable signaling in the face of promiscuity. These results suggest connections to both phase transitions in constraint satisfaction problems and coding theory bounds on the size of communication codes

Proteus: A Hierarchical Portfolio of Solvers and Transformations

In recent years, portfolio approaches to solving SAT problems and CSPs have become increasingly common. There are also a number of different encodings for representing CSPs as SAT instances. In this paper, we leverage advances in both SAT and CSP solving to present a novel hierarchical portfolio-based approach to CSP solving, which we call Proteus, that does not rely purely on CSP solvers. Instead, it may decide that it is best to encode a CSP problem instance into SAT, selecting an appropriate encoding and a corresponding SAT solver. Our experimental evaluation used an instance of Proteus that involved four CSP solvers, three SAT encodings, and six SAT solvers, evaluated on the most challenging problem instances from the CSP solver competitions, involving global and intensional constraints. We show that significant performance improvements can be achieved by Proteus obtained by exploiting alternative view-points and solvers for combinatorial problem-solving.Comment: 11th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. The final publication is available at link.springer.co

Quiet Planting in the Locked Constraint Satisfaction Problems

We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles. The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to be crucial. Our main result is the location of the hard region in the planted ensemble. In a part of that hard region instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio

Tricritical Points in Random Combinatorics: the (2+p)-SAT case

The (2+p)-Satisfiability (SAT) problem interpolates between different classes of complexity theory and is believed to be of basic interest in understanding the onset of typical case complexity in random combinatorics. In this paper, a tricritical point in the phase diagram of the random $2+p$-SAT problem is analytically computed using the replica approach and found to lie in the range $2/5 \le p_0 \le 0.416$. These bounds on $p_0$ are in agreement with previous numerical simulations and rigorous results.Comment: 7 pages, 1 figure, RevTeX, to appear in J.Phys.