A constraint programming (CP) solver that implements proof logging will output a machine-checkable certificate of correctness alongside any result it obtains. This is useful for trusting claims of unsatisfiability or optimality, as well as for debugging and auditing solver implementations. Proofs can be constructed by having the solver log justifications for each inference it makes, and previous work has shown that many standard CP reasoning techniques can be efficiently justified using a pseudo-Boolean (PB) proof format. This paper extends PB justifications to propagators enforcing bounds consistency on multiplication and division constraints. We show that even though the proof system and checker operate only on linear inequalities over 0-1 variables, non-linear reasoning over bounded domains can be efficiently expressed as a sequence of PB proof steps. Additionally, we demonstrate that bespoke proof logging for bounds-consistency algorithms offers a clear advantage over constructing justifications by brute force.

Pseudo-Boolean proof logging has been used successfully to provide certificates of optimality from a variety of constraint- and satisifability-style solvers that combine reasoning with a backtracking or clause-learning search. Another paradigm, occurring in dynamic programming and decision diagram solving, instead reasons about partial states and possible transitions between them. We describe a framework for generating clean and efficient pseudo-Boolean proofs for these kinds of algorithm, and use it to produce certifying algorithms for knapsack, longest path, and interval scheduling. Because we use a common proof system, we can also reason about hybrid solving algorithms: we demonstrate this by providing proof logging for a dynamic programming based knapsack propagator inside a constraint programming solver.

Proof logging in constraint programming is an approach to certifying a conclusion reached by a solver. To allow for this, different propagators must be augmented to produce justifications for any inferences they make, so that an independent proof checker can certify correctness. The Circuit constraint is used to enforce a Hamiltonian cycle on a set of vertices, e.g. for vehicle routing. Maintaining consistency for the Circuit constraint is hard, so various ad-hoc propagation techniques have been devised and implemented in solvers. We show that standard Circuit constraint inference rules can be efficiently justified within a pseudo-Boolean proof system, either by using a simple sequence of cutting planes steps or through a conditional counting argument.

It is well-known that variable ordering heuristics play a central role in solving efficiently Constraint Satisfaction Problem (CSP) instances. From the early 80’s, and during more than two decades, the dynamic variable ordering heuristic selecting the variable with the smallest domain was clearly prevailing. Then, from the mid 2000’s, some adaptive heuristics have been introduced: their principle is to collect some useful information during the search process in order to take better informed decisions. Among those adaptive heuristics, wdeg/dom (and its variants) remains particularly robust. In this paper, we introduce an original heuristic based on the midway processing of failing executions of constraint propagation: this heuristic called pick/dom tracks the variables that are directly involved in the process of constraint propagation, when ending with a conflict. The robustness of this new heuristic is demonstrated from a large experimentation conducted with the constraint solver ACE. Interestingly enough, one can observe some complementary between the early, midway and late forms of processing of conflicts.