Failure-based variable ordering heuristics (VOH) are efficient general-purpose search heuristics for solving constraint satisfaction problems (CSP). They learn from the failures detected during the search and select the variables that are most likely to fail. The current failure-based VOHs, i.e. the failure-rate-based (FRBA) and failure-length-based (FLBA), focus on only the failures detected in left branches. In this paper, we investigate how the failure information from right branches affects the performance of the failure-based VOHs. Four strategies utilizing the failure information of right branches are proposed to refine the failure-based VOHs. Our experiments performed with the benchmark instances used in the recent MiniZinc challenges show that utilizing the failures detected in right branches enhances the performance of the failure-based VOHs. The refined version combining all the proposed strategies generally gets the best performance. It demonstrates remarkable superiority over several general-purpose VOHs, including activity-based search, conflict-history search, refined weighted degree, pick/dom, and the existing FRBA, which are considered state-of-the-art. Our study demonstrates that right branches matter in failure-based VOHs.

The Bound-Impact Value Selector (BIVS) employs a look-ahead strategy to enable black-box Constraint Optimization Problem (COP) solvers to find high-quality solutions earlier. However, its computational cost prohibits its use throughout the entire search process. To mitigate this cost, the Restricted Fixpoint (RF) approach considers only the constraints on the shortest paths between the selected variable and the objective, yielding better performance. In this paper, we propose a lightweight strategy from a different perspective, named Assess Before Look-Ahead (ABLA), to enhance the performance of look-ahead-based value heuristics for solving COPs. ABLA first assesses whether the look-ahead process can differentiate between the values of a variable, and only performs the look-ahead when this assessment passes. Experiments on benchmark instances from recent MiniZinc Challenges demonstrate that ABLA’s decisions to skip redundant look-ahead processes are highly reliable, with an average accuracy of over 94%. Consequently, ABLA significantly boosts the performance of both BIVS and RF, outperforming two other baselines: the minimum value heuristic and the RLARF value heuristic.

Variable ordering heuristics (VOH) play a central role in solving Constraint Satisfaction Problems (CSP). The performance of different VOHs may vary greatly when solving the same CSP instance, so identifying an efficient candidate VOH for a given CSP has been a key issue in the community. In this study, we propose a prediction-based approach to adaptively select efficient VOHs for different CSPs from a set of candidates. Our work demonstrates that efficient candidate VOHs can be identified by learning from the topology of search trees. Specifically, we propose to represent the topology of a binary search tree by the sequence of the Numbers of Positive Decisions (NPD) made before each failure occurs. Based on the representation, we predict the total failure number of a search tree from its beginning part. When solving a CSP, we run a probing procedure to obtain the NPD sequences generated by candidate VOHs and select an efficient one for the resolution according to the prediction results. Our experiments show that the Long Short Term Memory model and Gradient Boosting Decision Tree models trained with the search trees sampled from easy instances are effective in identifying efficient VOHs for hard instances. The models capture some common structure properties hidden in the search trees of different problems. Our approach outperforms the state-of-the-art adaptive VOHs in terms of the number of solved instances and the PAR2 score of runtime.

AllDifferent constraint is widely used in Constraint Programming to model real world problems. Existing Generalized Arc Consistency (GAC) algorithms map an AllDifferent constraint onto a bipartite graph and utilize the structure of Strongly Connected Components (SCCs) in the graph to filter values. Calculating SCCs is time-consuming in the existing algorithms, so we propose a novel GAC algorithm for AllDifferent constraint in this paper, which eliminates the computation of SCCs. We prove that all redundant edges in the bipartite graph point to some alternating cycles. Our algorithm exploits this property and uses a more efficient method to filter values, which is based on breadth-first search. Experimental results on the XCSP3 benchmark suite show that our algorithm considerably outperforms the state-of-the-art GAC algorithms.

Variable ordering heuristics play a central role in solving constraint satisfaction problems. In this paper, we propose failure based variable ordering heuristics. Following the fail first principle, the new heuristics use two aspects of failure information collected during search. The failure rate heuristics consider the failure proportion after the propagations of assignments of variables and the failure length heuristics consider the length of failures, which is the number of fixed variables composing a failure. We performed a vast experiments in 41 problems with 1876 MiniZinc instances. The results show that the failure based heuristics outperform the existing ones including activity-based search, conflict history search, the refined weighted degree and correlation-based search. They can be new candidates of general purpose variable ordering heuristics for black-box CSP solvers.

Making good decisions at the top of a search tree is important for finding good solutions early in constraint optimization. In this paper, we propose a method employing frequent pattern mining (FPM), a classic datamining technique, to find good subtrees for solving constraint optimization problems. We demonstrate that applying FPM in a small number of random high-quality feasible solutions enables us to identify subtrees containing optimal solutions in more than 55% of problem instances for four real world benchmark problems. The method works as a plugin that can be combined with any search strategy for branch-and-bound search. Exploring the identified subtrees first, the method brings substantial improvements for four efficient search strategies in both total runtime and runtime of finding optimal solutions.

Simple Tabular Reduction algorithms (STR) work well to establish Generalized Arc Consistency (GAC) on positive table constraints. However, the existing STR algorithms are useless for negative table constraints. In this work, we propose a novel STR algorithm and its improvement, which work on negative table constraints. Our preliminary experiments are performed on some random instances and a certain benchmark instances. The results show that the new algorithms outperform GAC-valid and the MDD-based GAC algorithm.

Filtering techniques are essential in the search algorithms solving constraint satisfaction problems (CSPs). Arc consistency (AC) is the most often used filtering technique because it cheaply removes some values that cannot belong to any solutions. Comparing with AC, max-Restricted Path Consistency (maxRPC) has a stronger pruning power while it is not suited for use during search because of the prohibitive time cost. Thus, light maxRPC which is the light version of maxRPC was proposed. Comparing with maxRPC, it has a lower time cost and the search algorithm maintaining light maxRPC (MlmaxRPC) can outperform the search algorithm maintaining AC (MAC) on some problems. However, MlmaxRPC suffers from the time waste in the cases that applying a stricter checking standard does not intrigue any value deletion. In order to avoid the time waste in MlmaxRPC, in this paper, partial maxRPC which is a new approximation of maxRPC is proposed. It only applies the stricter checking standard when the value deletion is of high possibility. MpmaxRPC which is the search algorithm maintaining partial maxRPC has a better average performance than MAC and MlmaxRPC.