Incremental Cardinality Constraints for MaxSAT

Maximum Satisfiability (MaxSAT) is an optimization variant of the Boolean Satisfiability (SAT) problem. In general, MaxSAT algorithms perform a succession of SAT solver calls to reach an optimum solution making extensive use of cardinality constraints. Many of these algorithms are non-incremental in nature, i.e. at each iteration the formula is rebuilt and no knowledge is reused from one iteration to another. In this paper, we exploit the knowledge acquired across iterations using novel schemes to use cardinality constraints in an incremental fashion. We integrate these schemes with several MaxSAT algorithms. Our experimental results show a significant performance boost for these algo- rithms as compared to their non-incremental counterparts. These results suggest that incremental cardinality constraints could be beneficial for other constraint solving domains.Comment: 18 pages, 4 figures, 1 table. Final version published in Principles and Practice of Constraint Programming (CP) 201

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

On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: -- No algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the $O(m^{1/2-\varepsilon})$ bound for sparse graphs of Henzinger et al. [STOC'15] and $O(n^{1/3-\varepsilon})$ bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time $O(m\sqrt{n})$ exists for the former by Bosek et al. [FOCS'14]. -- No algorithm with amortized update time $O(m^{1-\varepsilon})$ exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time $O(n^{1/2 - \varepsilon})$ exists for incremental or decremental $(4/3-\varepsilon')$-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit

Exploiting Resolution-based Representations for MaxSAT Solving

Most recent MaxSAT algorithms rely on a succession of calls to a SAT solver in order to find an optimal solution. In particular, several algorithms take advantage of the ability of SAT solvers to identify unsatisfiable subformulas. Usually, these MaxSAT algorithms perform better when small unsatisfiable subformulas are found early. However, this is not the case in many problem instances, since the whole formula is given to the SAT solver in each call. In this paper, we propose to partition the MaxSAT formula using a resolution-based graph representation. Partitions are then iteratively joined by using a proximity measure extracted from the graph representation of the formula. The algorithm ends when only one partition remains and the optimal solution is found. Experimental results show that this new approach further enhances a state of the art MaxSAT solver to optimally solve a larger set of industrial problem instances