Constraints最新文献

We study a planning problem based on Plotting, a tile-matching puzzle video game published by Taito in 1989. The objective of this turn-based game is to remove a target number of coloured blocks from a grid by sequentially shooting blocks into the same grid. Plotting features complex transitions after every shot: various blocks are affected directly, while others can be indirectly affected by gravity. We consider modelling and solving Plotting from two perspectives. The puzzle is naturally cast as an AI Planning problem and we first discuss modelling the problem using the Planning Domain Definition Language (PDDL). We find that a model in which planning actions correspond to player actions is inefficient with a grounding-based state-of-the-art planner. However, with a more fine-grained action model, where each change of a block is a planning action, solving performance is dramatically improved. We also describe two lifted constraint models, able to capture the inherent complexities of Plotting and enabling the application of efficient solving approaches from SAT and CP. Our empirical results with these models demonstrates that they can compete with, and often exceed, the performance of the dedicated planning solvers, suggesting that the richer languages available to constraint modelling can be of benefit when considering planning problems with complex changes of state. CP and SAT solvers solved almost all of the largest and most challenging instances within 1 hour, whereas the best planning approach solved approximately 30%. Finally, the flexibility provided by the constraint models allows us to easily curate interesting levels for human players.

We develop and deploy a set of constraints for the purpose of calculating minimal sizes of lottery designs. Specifically, we find the minimum number of tickets of size six which are needed to match at least two balls on any draw of size six, whenever there are at most 70 balls.

We consider the problem of perception-based constraint solving, where part of the problem specification is provided indirectly through an image provided by a user. As a pedagogical example, we use the complete image of a Sudoku grid. While the rules of the puzzle are assumed to be known, the image must be interpreted by a neural network to extract the values in the grid. In this paper, we investigate (1) a hybrid modeling approach combining machine learning and constraint solving for joint inference, knowing that blank cells need to be both predicted as being blank and filled-in to obtain a full solution; (2) the effect of classifier calibration on joint inference; and (3) how to deal with cases where the constraints of the reasoning system are not satisfied. More specifically, in the case of handwritten user errors in the image, a naive approach fails to obtain a feasible solution even if the interpretation is correct. Our framework identifies human mistakes by using a constraint solver and helps the user to correct these mistakes. We evaluate the performance of the proposed techniques on images taken through the Sudoku Assistant Android app, among other datasets. Our experiments show that (1) joint inference can correct classifier mistakes, (2) overall calibration improves the solution quality on all datasets, and (3) estimating and discriminating between user-written and original visual input while reasoning makes for a more robust system, even in the presence of user errors.

Recently, mixed-integer programming (MIP) techniques have been applied to learn optimal decision trees. Empirical research has shown that optimal trees typically have better out-of-sample performance than heuristic approaches such as CART. However, the underlying MIP formulations often suffer from weak linear programming (LP) relaxations. Many existing MIP approaches employ big-M constraints to ensure observations are routed throughout the tree in a feasible manner. This paper introduces new MIP formulations for learning optimal decision trees with multivariate branching rules and no assumptions on the feature types. We first propose a strong baseline MIP formulation that still uses big-M constraints, but yields a stronger LP relaxation than its counterparts in the literature. We then introduce a problem-specific class of valid inequalities called shattering inequalities. Each inequality encodes an inclusion-minimal set of points that cannot be shattered by a multivariate split, and in the context of a MIP formulation, the inequalities are sparse, involving at most the number of features plus two variables. We propose a separation procedure that attempts to find a violated inequality given a (possibly fractional) solution to the LP relaxation; in the case where the solution is integer, the separation is exact. Numerical experiments show that our MIP approach outperforms two other MIP formulations in terms of solution time and relative gap, and is able to improve solution time while remaining competitive with regards to out-of-sample accuracy in comparison to a wider range of approaches from the literature.

t-wise coverage is one of the most important techniques used to test configurations of software for finding bugs. It ensures that interactions between features of a Software Product Line (SPL) are tested. The size of SPLs (of thousands of features) makes the problem of finding a good test suite computationally expensive, as the number of t-wise combinations grows exponentially. In this article, we leverage Constraint Programming’s search strategies to generate test suites with a high coverage of configurations. We analyse the behaviour of the default random search strategy, and then we propose an improvement based on the commonalities (frequency) of the features. We experimentally compare to uniform sampling and state of the art sampling approaches. We show that our new search strategy outperforms all the other approaches and has the fastest running time.