Optimization Modulo Theories with Linear Rational Costs

In the contexts of automated reasoning (AR) and formal verification (FV), important decision problems are effectively encoded into Satisfiability Modulo Theories (SMT). In the last decade efficient SMT solvers have been developed for several theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors). Surprisingly, little work has been done to extend SMT to deal with optimization problems; in particular, we are not aware of any previous work on SMT solvers able to produce solutions which minimize cost functions over arithmetical variables. This is unfortunate, since some problems of interest require this functionality. In the work described in this paper we start filling this gap. We present and discuss two general procedures for leveraging SMT to handle the minimization of linear rational cost functions, combining SMT with standard minimization techniques. We have implemented the procedures within the MathSAT SMT solver. Due to the absence of competitors in the AR, FV and SMT domains, we have experimentally evaluated our implementation against state-of-the-art tools for the domain of linear generalized disjunctive programming (LGDP), which is closest in spirit to our domain, on sets of problems which have been previously proposed as benchmarks for the latter tools. The results show that our tool is very competitive with, and often outperforms, these tools on these problems, clearly demonstrating the potential of the approach.Comment: Submitted on january 2014 to ACM Transactions on Computational Logic, currently under revision. arXiv admin note: text overlap with arXiv:1202.140

Planning as Quantified Boolean Formulae

This work explores the idea of classical Planning as Quantified Boolean Formulae. Planning as Satisfiability (SAT) is a popular approach to Planning and has been explored in detail producing many compact and efficient encodings, Planning-specific solver implementations and innovative new constraints. However, Planning as Quantified Boolean Formulae (QBF) has been relegated to conformant Planning approaches, with the exception of one encoding that has not yet been investigated in detail. QBF is a promising setting for Planning given that the problems have the same complexity. This work introduces two approaches for translating bounded propositional reachability problems into QBF. Both exploit the expressivity of the binarytree structure of the QBF problem to produce encodings that are as small as logarithmic in the size of the instance and thus exponentially smaller than the corresponding SAT encoding with the same bound. The first approach builds on the iterative squaring formulation of Rintanen; the intuition behind the idea is to recursively fold the plan around the midpoint, reducing the number of time-steps that need to be described from n to log₂n. The second approach exploits domain-level lifting to achieve significant improvements in efficiency. Experimentation was performed to compare our formulation of the first approach with the previous formulation, and to compare both approaches with comparative and state-of-the-art SAT approaches. Results presented in this work show that our formulation of the first approach is an improvement over the previous, and that both approaches produce encodings that are indeed much smaller than corresponding SAT encodings, in both terms of encoding size and memory used during solving. Evidence is also provided to show that the first approach is feasible, if not yet competitive with the state-of-the-art, and that the second approach produces superior encodings to the SAT encodings when the domain is suited to domain-level lifting.This work explores the idea of classical Planning as Quantified Boolean Formulae. Planning as Satisfiability (SAT) is a popular approach to Planning and has been explored in detail producing many compact and efficient encodings, Planning-specific solver implementations and innovative new constraints. However, Planning as Quantified Boolean Formulae (QBF) has been relegated to conformant Planning approaches, with the exception of one encoding that has not yet been investigated in detail. QBF is a promising setting for Planning given that the problems have the same complexity. This work introduces two approaches for translating bounded propositional reachability problems into QBF. Both exploit the expressivity of the binarytree structure of the QBF problem to produce encodings that are as small as logarithmic in the size of the instance and thus exponentially smaller than the corresponding SAT encoding with the same bound. The first approach builds on the iterative squaring formulation of Rintanen; the intuition behind the idea is to recursively fold the plan around the midpoint, reducing the number of time-steps that need to be described from n to log₂n. The second approach exploits domain-level lifting to achieve significant improvements in efficiency. Experimentation was performed to compare our formulation of the first approach with the previous formulation, and to compare both approaches with comparative and state-of-the-art SAT approaches. Results presented in this work show that our formulation of the first approach is an improvement over the previous, and that both approaches produce encodings that are indeed much smaller than corresponding SAT encodings, in both terms of encoding size and memory used during solving. Evidence is also provided to show that the first approach is feasible, if not yet competitive with the state-of-the-art, and that the second approach produces superior encodings to the SAT encodings when the domain is suited to domain-level lifting

Scheduling stretched coupled-tasks with compatibilities constraints : model, complexity and approximation results for some class of graphs

We tackle the makespan minimization coupled-tasks problem in presence of compatibility constraints. In particular, we focus on stretched coupled-tasks, {\it i.e.}coupled-tasks having the same sub-tasks execution time and idle time duration. We study severals problems in frame works of classic complexity and approximation for which the compatibility graph $G_c$ is bipartite (star, chain, $\ldots$) In such context, we design some efficient polynomial-time approximation algorithms according to difference parameters of the scheduling problem. When $G_c$ is a $k$-stage bipartite graph, we propose, among other, a $\frac{7}{6}$-approximation algorithm when $k=1$, and a $\frac{13}{9}$-approximation algorithm when $k=2$.\

Efficient Automated Planning with New Formulations

Problem solving usually strongly relies on how the problem is formulated. This fact also applies to automated planning, a key field in artificial intelligence research. Classical planning used to be dominated by STRIPS formulation, a simple model based on propositional logic. In the recently introduced SAS+ formulation, the multi-valued variables naturally depict certain invariants that are missed in STRIPS, make SAS+ have many favorable features. Because of its rich structural information SAS+ begins to attract lots of research interest. Existing works, however, are mostly limited to one single thing: to improve heuristic functions. This is in sharp contrast with the abundance of planning models and techniques in the field. On the other hand, although heuristic is a key part for search, its effectiveness is limited. Recent investigations have shown that even if we have almost perfect heuristics, the number of states to visit is still exponential. Therefore, there is a barrier between the nice features of SAS+ and its applications in planning algorithms. In this dissertation, we have recasted two major planning paradigms: state space search and planning as Satisfiability: SAT), with three major contributions. First, we have utilized SAS+ for a new hierarchical state space search model by taking advantage of the decomposable structure within SAS+. This algorithm can greatly reduce the time complexity for planning. Second, planning as Satisfiability is a major planning approach, but it is traditionally based on STRIPS. We have developed a new SAS+ based SAT encoding scheme: SASE) for planning. The state space modeled by SASE shows a decomposable structure with certain components independent to others, showing promising structure that STRIPS based encoding does not have. Third, the expressiveness of planning is important for real world scenarios, thus we have also extended the planning as SAT to temporally expressive planning and planning with action costs, two advanced features beyond classical planning. The resulting planner is competitive to state-of-the-art planners, in terms of both quality and performance. Overall, our work strongly suggests a shifting trend of planning from STRIPS to SAS+, and shows the power of formulating planning problems as Satisfiability. Given the important roles of both classical planning and temporal planning, our work will inspire new developments in other advanced planning problem domains

Efficient satisfiability solver

The past few decades saw great improvements in the performance of satisfiability (SAT) solvers. In this thesis, we discuss the state-of-the-art techniques used in building an efficient SAT solver. Modern SAT solvers are mainly constituted by the following components: decision heuristics, Boolean constraint propagation, conflict analysis, restart, clause deletion and preprocessing. Various algorithms and implementations in each component will be discussed and analyzed. Then we propose a new backtracking strategy, partial backtracking, which can be easily implemented in SAT solvers. It is essentially an extension of the backtracking strategy used in most SAT solvers. With partial backtracking, the solver consecutively amends the variable assignments instead of discarding them completely so that it does not backtrack as many levels as the classic strategy does after analyzing a conflict. We implemented this strategy in our solver Nigma and the experiments show that the solver benefits from this adjustment