Abstract MaxSAT modulo theories (MaxSMT) is an important generalization of Satisfiability modulo theories (SMT) with various applications. In this paper, we focus on MaxSMT with the background theory of Linear Integer Arithmetic, denoted as MaxSMT(LIA). We design the first local search algorithm for MaxSMT(LIA) called PairLS, based on the following novel ideas. A novel operator called pairwise operator is proposed for integer variables. It extends the original local search operator by simultaneously operating on two variables, enriching the search space. Moreover, a compensation-based picking heuristic is proposed to determine and distinguish the pairwise operations. Experiments are conducted to evaluate our algorithm on massive benchmarks. The results show that our solver is competitive with state-of-the-art MaxSMT solvers. Furthermore, we also apply the pairwise operation to enhance the local search algorithm of SMT, which shows its extensibility.

Satisfiability Modulo Theory (SMT) generalizes the propositional satisfiability problem (SAT) by extending support for various first-order background theories. In this paper, we focus on the SMT problems in Non-Linear Integer Arithmetic (NIA) theory, referred to as SMT(NIA), which has wide applications in software engineering. The dominant paradigm for SMT(NIA) is the CDCL(T) framework, while recently stochastic local search (SLS) has also shown its effectiveness. However, the cooperation between the two methods has not been studied yet. Motivated by the great success of the deep cooperation of CDCL and SLS for SAT, we propose a two-layer hybrid approach for SMT(NIA). The outer-layer interleaves between the inner-layer and an independent SLS solver. In the inner-layer, we take CDCL(T) as the main body, and design DCL(T)-guided SLS solver, which is invoked at branches corresponding to skeleton solutions and returns useful information to improve the branching heuristics of CDCL(T). We implement our ideas on top of the CDCL(T) tactic of Z3 with an SLS solver called LocalSMT, resulting in a hybrid solver dubbed HybridSMT. Extensive experiments are carried out on the standard SMT(NIA) benchmarks from SMT-LIB, where most of the instances are from real-world software engineering applications of termination and non-termination analysis. Experiment results show that HybridSMT significantly improves the CDCL(T) solver in Z3. Moreover, our solver can solve 10.36% more instances than the currently best SMT(NIA) solver, and is more efficient for software verification instances.

Satisfiability Modulo Theories (SMT) refers to the problem of deciding the satisfiability of a formula with respect to certain background first-order theories. In this article, we focus on Satisfiablity Modulo Integer Arithmetic, which is referred to as SMT(IA), including both linear and non-linear integer arithmetic theories. Dominant approaches to SMT rely on calling a CDCL-based SAT solver, either in a lazy or eager flavour. Local search, a competitive approach to solving combinatorial problems including SAT, however, has not been well studied for SMT. We develop the first local-search algorithm for SMT(IA) by directly operating on variables, breaking through the traditional framework. We propose a local-search framework by considering the distinctions between Boolean and integer variables. Moreover, we design a novel operator and scoring functions tailored for integer arithmetic, as well as a two-level operation selection heuristic. Putting these together, we develop a local search SMT(IA) solver called LocalSMT. Experiments are carried out to evaluate LocalSMT on benchmark sets from SMT-LIB. The results show that LocalSMT is competitive and complementary with state-of-the-art SMT solvers, and performs particularly well on those formulae with only integer variables. A simple sequential portfolio with Z3 improves the state-of-the-art on satisfiable benchmark sets from SMT-LIB.

Abstract Satisfiability Modulo Linear Integer Arithmetic, SMT (LIA) for short, has significant applications in many domains. In this paper, we develop the first local search algorithm for SMT (LIA) by directly operating on variables, breaking through the traditional framework. We propose a local search framework by considering the distinctions between Boolean and integer variables. Moreover, we design a novel operator and scoring functions tailored for LIA, and propose a two-level operation selection heuristic. Putting these together, we develop a local search SMT (LIA) solver called LS-LIA. Experiments are carried out to evaluate LS-LIA on benchmarks from SMTLIB and two benchmark sets generated from job shop scheduling and data race detection. The results show that LS-LIA is competitive and complementary with state-of-the-art SMT solvers, and performs particularly well on those formulae with only integer variables. A simple sequential portfolio with Z3 improves the state-of-the-art on satisfiable benchmark sets of LIA and IDL benchmarks from SMT-LIB. LS-LIA also solves Job Shop Scheduling benchmarks substantially faster than traditional complete SMT solvers.

The minimum weighted connected dominating set (MWCDS) problem is an important variant of connected dominating set problems with wide applications, especially in heterogenous networks and gene regulatory networks. In the paper, we develop a nested local search algorithm called NestedLS for solving MWCDS on classic benchmarks and massive graphs. In this local search framework, we propose two novel ideas to make it effective by utilizing previous search information. First, we design the restart based smoothing mechanism as a diversification method to escape from local optimal. Second, we propose a novel inner-layer local search method to enlarge the candidate removal set, which can be modelled as an optimized version of spanning tree problem. Moreover, inner-layer local search method is a general method for maintaining the connectivity constraint when dealing with massive graphs. Experimental results show that NestedLS outperforms state-of-the-art meta-heuristic algorithms on most instances.

The minimum connected dominating set (MCDS) problem is an important extension of the minimum dominating set problem, with wide applications, especially in wireless networks. Most previous works focused on solving MCDS problem in graphs with relatively small size, mainly due to the complexity of maintaining connectivity. This paper explores techniques for solving MCDS problem in massive real-world graphs with wide practical importance. Firstly, we propose a local greedy construction method with reasoning rule called 1hopReason. Secondly and most importantly, a hybrid dynamic connectivity maintenance method (HDC+) is designed to switch alternately between a novel fast connectivity maintenance method based on spanning tree and its previous counterpart. Thirdly, we adopt a two-level vertex selection heuristic with a newly proposed scoring function called chronosafety to make the algorithm more considerate when selecting vertices. We design a new local search algorithm called FastCDS based on the three ideas. Experiments show that FastCDS significantly outperforms five state-of-the-art MCDS algorithms on both massive graphs and classic benchmarks.

The minimum connected dominating set (MCDS) problem is an important extension of the minimum dominating set problem, with wide applications, especially in wireless networks. Despite its practical importance, there are few works on solving MCDS for massive graphs, mainly due to the complexity of maintaining connectivity. In this paper, we propose two novel ideas, and develop a new local search algorithm for MCDS called NuCDS. First, a hybrid dynamic connectivity maintenance method is designed to switch alternately between a novel fast connectivity maintenance method based on spanning tree and its previous counterpart. Second, we define a new vertex property called \emph{safety} to make the algorithm more considerate when selecting vertices. Experiments show that NuCDS significantly outperforms the state-of-the-art MCDS algorithms on both massive graphs and classic benchmarks.