Design, implementation and evaluation of a distributed CDCL framework

The primary subject of this dissertation is practically solving instances of the Boolean satisfiability problem (SAT) that arise from industrial applications. The invention of the conflict-driven clause-learning (CDCL) algorithm led to enormous progress in this field. CDCL has been augmented with effective pre- and inprocessing techniques that boost its effectiveness. While a considerable amount of work has been done on applying shared-memory parallelism to enhance the performance of CDCL, solving SAT on distributed architectures is studied less thoroughly. In this work, we develop a distributed, CDCL-based framework for SAT solving. This framework consists of three main components: 1. An implementation of the CDCL algorithm that we have written from scratch, 2. a novel, parallel SAT algorithm that builds upon this CDCL implementation and 3. a collection of parallel simplification techniques for SAT instances. We call our resulting framework satUZK; our parallel solving algorithm is called the distributed divide-and-conquer (DDC) algorithm. The DDC algorithm employs a parallel lookahead procedure to dynamically partition the search space. Load balancing is used to ensure that all computational resources are utilized during lookahead. This procedure results in a divide-and-conquer tree that is distributed over all processors. Individual threads are routed through this tree until they arrive at unsolved leaf vertices. Upon arrival, the lookahead procedure is invoked again or the leaf vertex is solved via CDCL. Several extensions to the DDC algorithm are proposed. These include clause sharing and a scheme to locally adjust the LBD score relative to the current search tree vertex. LBD is a measure for the usefulness of clauses that participate in a CDCL search. We evaluate our DDC algorithm empirically and benchmark it against the best distributed SAT algorithms. In this experiment, our DDC algorithm is faster than other distributed, state-of-the-art solvers and solves at least as many instances. In addition to running a parallel algorithm for SAT solving we also consider parallel simplifcation. Here, we first develop a theoretical foundation that allows us to prove the correctness of parallel simplification techniques. Using this as a basis, we examine established simplification algorithms for their parallelizability. It turns out that several well-known simplification techniques can be parallelized efficiently. We provide parallel implementation of the techniques and test their effectiveness in empirical experiments. This evaluation finds several combinations of simplification techniques that can solve instances which could not be solved by the DDC algorithm alone

Satisfiability Test with Synchronous Simulated Annealing on the Fujitsu AP1000 Massively-Parallel Multiprocessor

Solving the hard Satisfiability Problem is time consuming even for modest-sized problem instances. Solving the Random L-SAT Problem is especially difficult due to the ratio of clauses to variables. This report presents a parallel synchronous simulated annealing method for solving the Random L-SAT Problem on a large-scale distributed-memory multiprocessor. In particular, we use a parallel synchronous simulated annealing procedure, called Generalized Speculative Computation, which guarantees the same decision sequence as sequential simulated annealing. To demonstrate the performance of the parallel method, we have selected problem instances varying in size from 100-variables/425-clauses to 5000-variables/21,250-clauses. Experimental results on the AP1000 multiprocessor indicate that our approach can satisfy 99.9 percent of the clauses while giving almost a 70-fold speedup on 500 processors

Grid based propositional satisfiability solving

This work studies how grid and cloud computing can be applied to efficiently solving propositional satisfiability problem (SAT) instances. Propositional logic provides a convenient language for expressing real-world originated problems such as AI planning, automated test pattern generation, bounded model checking and cryptanalysis. The interest in SAT solving has increased mainly due to improvements in the solving algorithms, which recently have increasingly focused on using parallelism offered by multi-CPU computers. Partly orthogonally to these improvements this work studies several novel approaches to parallel solving of SAT instances in a grid of widely distributed "virtual" computers instead of workstations or supercomputers. Two types of parallel SAT solving approaches are analyzed and used as building blocks for more complex systems: using several solvers which compete to solve a given instance in parallel, and splitting the search space of the instance and solving the resulting partitions in parallel. The work presents several efficient partitioning functions, critical in successful splitting according to an analytical result, and presents novel solving systems that are less dependent on the partitioning function efficiency. Finally, the work studies combining clause learning, a key technique in modern SAT solvers, with the novel types of parallel solvers. Different heuristics are studied for filtering clauses learned in parallel, and the work proposes techniques which allow exchanging the clauses between different splits. The practical significance of the results are studied using large, standard benchmark sets from SAT competitions. Some of the approaches are able to solve several instances that have either not been solved at all by any other solver, or which are significantly slower to solve with other solvers

Distributed Domain Propagation

This is the final version. Available on open access from the publisher via the DOI in this record16th International Symposium on Experimental Algorithms (SEA 2017), 21-23 June 2017, London, UKPortfolio parallelization is an approach that runs several solver instances in parallel and terminates when one of them succeeds in solving the problem. Despite it’s simplicity portfolio parallelization has been shown to perform well for modern mixed-integer programming (MIP) and boolean satisfiability problem (SAT) solvers. Domain propagation has also been shown to be a simple technique in modern MIP and SAT solvers that effectively finds additional domain reductions after a variables domain has been reduced. This paper investigates the impact of distributed domain propagation in modern MIP solvers that employ portfolio parallelization. Computational experiments were conducted for two implementations of this parallelization approach. While both share global variable bounds and solutions they communicate differently. In one implementation the communication is performed only at designated points in the solving process and in the other it is performed completely asynchronously. Computational experiments show a positive performance impact of communicating global variable bounds and provide valuable insights in communication strategies for parallel solvers.German Federal Ministry of Education and Researc

Tarmo: A Framework for Parallelized Bounded Model Checking

This paper investigates approaches to parallelizing Bounded Model Checking (BMC) for shared memory environments as well as for clusters of workstations. We present a generic framework for parallelized BMC named Tarmo. Our framework can be used with any incremental SAT encoding for BMC but for the results in this paper we use only the current state-of-the-art encoding for full PLTL. Using this encoding allows us to check both safety and liveness properties, contrary to an earlier work on distributing BMC that is limited to safety properties only. Despite our focus on BMC after it has been translated to SAT, existing distributed SAT solvers are not well suited for our application. This is because solving a BMC problem is not solving a set of independent SAT instances but rather involves solving multiple related SAT instances, encoded incrementally, where the satisfiability of each instance corresponds to the existence of a counterexample of a specific length. Our framework includes a generic architecture for a shared clause database that allows easy clause sharing between SAT solver threads solving various such instances. We present extensive experimental results obtained with multiple variants of our Tarmo implementation. Our shared memory variants have a significantly better performance than conventional single threaded approaches, which is a result that many users can benefit from as multi-core and multi-processor technology is widely available. Furthermore we demonstrate that our framework can be deployed in a typical cluster of workstations, where several multi-core machines are connected by a network

Distributed Random Convex Programming via Constraints Consensus

This paper discusses distributed approaches for the solution of random convex programs (RCP). RCPs are convex optimization problems with a (usually large) number N of randomly extracted constraints; they arise in several applicative areas, especially in the context of decision under uncertainty, see [2],[3]. We here consider a setup in which instances of the random constraints (the scenario) are not held by a single centralized processing unit, but are distributed among different nodes of a network. Each node "sees" only a small subset of the constraints, and may communicate with neighbors. The objective is to make all nodes converge to the same solution as the centralized RCP problem. To this end, we develop two distributed algorithms that are variants of the constraints consensus algorithm [4],[5]: the active constraints consensus (ACC) algorithm, and the vertex constraints consensus (VCC) algorithm. We show that the ACC algorithm computes the overall optimal solution in finite time, and with almost surely bounded communication at each iteration. The VCC algorithm is instead tailored for the special case in which the constraint functions are convex also w.r.t. the uncertain parameters, and it computes the solution in a number of iterations bounded by the diameter of the communication graph. We further devise a variant of the VCC algorithm, namely quantized vertex constraints consensus (qVCC), to cope with the case in which communication bandwidth among processors is bounded. We discuss several applications of the proposed distributed techniques, including estimation, classification, and random model predictive control, and we present a numerical analysis of the performance of the proposed methods. As a complementary numerical result, we show that the parallel computation of the scenario solution using ACC algorithm significantly outperforms its centralized equivalent

Survey-propagation decimation through distributed local computations

We discuss the implementation of two distributed solvers of the random K-SAT problem, based on some development of the recently introduced survey-propagation (SP) algorithm. The first solver, called the "SP diffusion algorithm", diffuses as dynamical information the maximum bias over the system, so that variable nodes can decide to freeze in a self-organized way, each variable making its decision on the basis of purely local information. The second solver, called the "SP reinforcement algorithm", makes use of time-dependent external forcing messages on each variable, which let the variables get completely polarized in the direction of a solution at the end of a single convergence. Both methods allow us to find a solution of the random 3-SAT problem in a range of parameters comparable with the best previously described serialized solvers. The simulated time of convergence towards a solution (if these solvers were implemented on a distributed device) grows as log(N).Comment: 18 pages, 10 figure