Integer polyhedra for program analysis

Polyhedra are widely used in model checking and abstract interpretation. Polyhedral analysis is effective when the relationships between variables are linear, but suffers from imprecision when it is necessary to take into account the integrality of the represented space. Imprecision also arises when non-linear constraints occur. Moreover, in terms of tractability, even a space defined by linear constraints can become unmanageable owing to the excessive number of inequalities. Thus it is useful to identify those inequalities whose omission has least impact on the represented space. This paper shows how these issues can be addressed in a novel way by growing the integer hull of the space and approximating the number of integral points within a bounded polyhedron

Delta-Decision Procedures for Exists-Forall Problems over the Reals

Solving nonlinear SMT problems over real numbers has wide applications in robotics and AI. While significant progress is made in solving quantifier-free SMT formulas in the domain, quantified formulas have been much less investigated. We propose the first delta-complete algorithm for solving satisfiability of nonlinear SMT over real numbers with universal quantification and a wide range of nonlinear functions. Our methods combine ideas from counterexample-guided synthesis, interval constraint propagation, and local optimization. In particular, we show how special care is required in handling the interleaving of numerical and symbolic reasoning to ensure delta-completeness. In experiments, we show that the proposed algorithms can handle many new problems beyond the reach of existing SMT solvers

Lazy clause generation reengineered

Abstract. Lazy clause generation is a powerful hybrid approach to combinatorial optimization that combines features from SAT solving and finite domain (FD) propagation. In lazy clause generation finite domain propagators are considered as clause generators that create a SAT description of their behaviour for a SAT solver. The ability of the SAT solver to explain and record failure and perform conflict directed backjumping are then applicable to FD problems. The original implementation of lazy clause generation was constructed as a cut down finite domain propagation engine inside a SAT solver. In this paper we show how to engineer a lazy clause generation solver by embedding a SAT solver inside an FD solver. The resulting solver is flexible, efficient and easy to use. We give experiments illustrating the effect of different design choices in engineering the solver.

A hybrid constraint model for the routing and wavelength assignment problem

Abstract. In this paper we present a hybrid model for the demand acceptance variant of the routing and wavelength assignment problem in directed networks, an important benchmark problem in optical network design. Our solution uses a decomposition into a MIP model for the routing and optimization aspect, combined with a finite domain constraint model for the wavelength assignment. If a solution to the constraint problem is found, it provides an optimal solution to the overall problem. If the constraint problem is infeasible, we use an extended explanation technique to find a good relaxation of the problem which leads to a near optimal solution. Extensive experiments show that proven optimality is achieved for more than 99.8 % of all cases tested, while run-times are orders of magnitude smaller than the best known MIP solution.