Since their original inception to represent Boolean functions for verification problems in the 1950s, decision diagrams have found wide applicability across academic disciplines and industry. This presentation discusses the use of decision diagrams as a compact representation of feasible solutions to constrained optimization problems, where solutions correspond to paths in a layered graph. By using relaxed and restricted decision diagrams of bounded size, one can balance the strength of the representation and computational effort. We highlight three roles of decision diagrams in constrained optimization. First, they enable a model-and-solve approach for dynamic programming, where a dynamic programming model and a merging rule define the compilation of decision diagrams that yield primal and dual bounds within a state-based search. Second, in constraint programming, they strengthen constraint propagation through multi-valued decision diagrams and provide optimization bounds within the search process. Third, in integer programming, they yield arc-flow formulations and establish connections with Dantzig–Wolfe decomposition, leading to strong bounds and state-of-the-art computational results. These approaches are illustrated on applications including machine scheduling, graph multi-coloring, and vehicle routing, where decision diagram-based methods have led to substantial improvements on benchmark instances. They have also been adopted in practice, both as a dual bounding component within a general-purpose optimization solver and in industrial applications for routing and scheduling.

Branch-and-bound methods for combinatorial optimization rely critically on the efficient computation of strong bounds during search. Decision diagram–based optimization provides such bounds via restricted and relaxed multi-valued decision diagrams (MDDs), but compiling relaxed diagrams can become a computational bottleneck for existing solvers. We present a GPU-accelerated implementation of decision diagram–based branch-and-bound using a decoupled architecture. It separates the compilation of relaxed and restricted diagrams and coordinates them through two queues of search states. This design enables heterogeneous parallelization: restricted diagrams are compiled concurrently on CPU threads while relaxed diagrams are constructed in parallel on a GPU. The GPU implementation exploits the layered structure of decision diagrams by expanding states in parallel and performing successor generation, dominance filtering, and state merging on the GPU. Computational experiments on knapsack, maximum independent set, and Golomb ruler benchmarks demonstrate substantial performance improvements over CPU-based decision diagram solvers, including speedups of up to an order of magnitude on hard instances and the ability to solve Golomb ruler instances up to size 16.

We introduce CODD, a system for solving combinatorial optimization problems using decision diagram technology. Problems are represented as state-based dynamic programming models using the CODD language specification. The model specification is used to automatically compile relaxed and restricted decision diagrams that are embedded inside a branch-and-bound search process. We introduce abstractions that allow us to generically implement the solver components while maintaining overall execution efficiency. We demonstrate the functionality of CODD on a variety of combinatorial optimization problems and compare its performance to other state-based solvers as well as integer programming and constraint programming solvers. CODD provides competitive results and can outperform the other solvers, sometimes by orders of magnitude.

This paper develops a memory-efficient approach for Sequential Pattern Mining (SPM), a fundamental topic in knowledge discovery that faces a well-known memory bottleneck for large data sets. Our methodology involves a novel hybrid trie data structure that exploits recurring patterns to compactly store the data set in memory; and a corresponding mining algorithm designed to effectively extract patterns from this compact representation. Numerical results on small to medium-sized real-life test instances show an average improvement of 85% in memory consumption and 49% in computation time compared to the state of the art. For large data sets, our algorithm stands out as the only capable SPM approach within 256GB of system memory, potentially saving 1.7TB in memory consumption.

Pattern mining is an essential part of knowledge discovery and data analytics. It is a powerful paradigm, especially when combined with constraint reasoning. In this paper, we present Seq2Pat, a constraint-based sequential pattern mining tool with a high-level declarative user interface. The library finds patterns that frequently occur in large sequence databases subject to constraints. We highlight key benefits that are desirable, especially in industrial settings where scalability, explainability, rapid experimentation, reusability, and reproducibility are of great interest. We then showcase an automated feature extraction process powered by Seq2Pat to discover high-level insights and boost downstream machine learning models for customer intent prediction.

Many real-world structured prediction problems need machine learning to capture data distribution and constraint reasoning to ensure structure validity. Nevertheless, constrained structured prediction is still limited in real-world applications because of the lack of tools to bridge constraint satisfaction and machine learning. In this paper, we propose COnstraint REasoning embedded Structured Prediction (Core-Sp), a scalable constraint reasoning and machine learning integrated approach for learning over structured domains. We propose to embed decision diagrams, a popular constraint reasoning tool, as a fully-differentiable module into deep neural networks for structured prediction. We also propose an iterative search algorithm to automate the searching process of the best Core-Sp structure. We evaluate Core-Sp on three applications: vehicle dispatching service planning, if-then program synthesis, and text2SQL generation. The proposed Core-Sp module demonstrates superior performance over state-of-the-art approaches in all three applications. The structures generated with Core-Sp satisfy 100% of the constraints when using exact decision diagrams. In addition, Core-Sp boosts learning performance by reducing the modeling space via constraint satisfaction.

This paper develops an exact solution algorithm for the multiple sequence alignment (MSA) problem. In the first step, we design a dynamic programming model and use it to construct a novel multivalued decision diagram (MDD) representation of all pairwise sequence alignments (PSA). PSA MDDs are then synchronized using side constraints to model the MSA problem as a mixed-integer program (MIP), for the first time, in polynomial space complexity. Two bound-based filtering procedures are developed to reduce the size of the MDDs, and the resulting MIP is solved using logic-based Benders decomposition. For a more effective algorithm, we develop a two-phase solution approach. In the first phase, we use optimistic filtering to quickly obtain a near-optimal bound, which we then use for exact filtering in the second phase to prove or obtain an optimal solution. Numerical results on benchmark instances show that our algorithm solves several instances to optimality for the first time, and, in case optimality cannot be proven, considerably improves upon a state-of-the-art heuristic MSA solver. Comparison with an existing state-of-the-art exact MSA algorithm shows that our approach is more time efficient and yields significantly smaller optimality gaps.

Constraint-based sequential pattern mining aims at identifying frequent patterns on a sequential database of items while observing constraints defined over the item attributes. We introduce novel techniques for constraint-based sequential pattern mining that rely on a multi-valued decision diagram (MDD) representation of the database. Specifically, our representation can accommodate multiple item attributes and various constraint types, including a number of non-monotone constraints. To evaluate the applicability of our approach, we develop an MDD-based prefix-projection algorithm and compare its performance against a typical generate-and-check variant, as well as a state-of-the-art constraint-based sequential pattern mining algorithm. Results show that our approach is competitive with or superior to these other methods in terms of scalability and efficiency.

The most common approach to generate cuts in integer programming is to derive them from the linear programming relaxation. We study an alternative approach that extracts cuts from discrete relaxations known as relaxed decision diagrams. Through a connection between decision diagrams and polarity, the algorithm generates cuts that are facet defining for the convex hull of a decision diagram relaxation. As proof of concept, we provide computational evidence that this algorithm generates strong cuts for the maximum independent set problem and the minimum set covering problem. The online appendices are available at https://doi.org/10.1287/ijoc.2018.0830 .

We propose a general branch-and-bound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, and restricted BDDs supply a primal heuristic. Each problem is given a dynamic programming model that allows one to exploit recursive structure, even though the problem is not solved by dynamic programming. A novel search scheme branches within relaxed BDDs rather than on values of variables. Preliminary testing shows that a rudimentary BDD-based solver is competitive with or superior to a leading commercial integer programming solver for the maximum stable set problem, the maximum cut problem on a graph, and the maximum 2-satisfiability problem. Specific to the maximum cut problem, we tested the BDD-based solver on a classical benchmark set and identified tighter relaxation bounds than have ever been found by any technique, nearly closing the entire optimality gap on four large-scale instances.

We explore the idea of obtaining bounds on the value of an optimization problem from a discrete relaxation based on binary decision diagrams (BDDs). We show how to construct a BDD that represents a relaxation of a 0-1 optimization problem, and how to obtain a bound for a separable objective function by solving a shortest (or longest) path problem in the BDD. As a test case we apply the method to the maximum independent set problem on a graph. We find that for most problem instances, it delivers tighter bounds in less computation time, than state-of-the-art integer programming software obtains by solving a continuous relaxation augmented with cutting planes.

We study propagation for the Sequence constraint in the context of constraint programming based on limited-width MDDs. Our first contribution is proving that establishing MDD-consistency for Sequence is NP-hard. Yet, we also show that this task is fixed parameter tractable with respect to the length of the sub-sequences. In addition, we propose a partial filtering algorithm that relies on a specific decomposition of the constraint and a novel extension of MDD filtering to node domains. We experimentally evaluate the performance of our proposed filtering algorithm, and demonstrate that the strength of the MDD propagation increases as the maximum width is increased. In particular, MDD propagation can outperform conventional domain propagation for Sequence by reducing the search tree size and solving time by several orders of magnitude. Similar improvements are observed with respect to the current best MDD approach that applies the decomposition of Sequence into Among constraints.