The branch-and-bound algorithm based on decision diagrams is a framework for solving discrete optimization problems with a dynamic programming formulation. It works by compiling a series of bounded-width decision diagrams that can provide lower and upper bounds for any given subproblem. Eventually, every part of the search space will be either explored or pruned by the algorithm, thus proving optimality. This paper presents new ingredients to speed up the search by exploiting the structure of dynamic programming models. The key idea is to prevent the repeated expansion of nodes corresponding to the same dynamic programming states by querying expansion thresholds cached throughout the search. These thresholds are based on dominance relations between partial solutions previously found and on pruning inequalities given by rough upper bounds and local bounds — two additional filtering techniques recently introduced. Computational experiments show that the pruning brought by this caching mechanism allows for significantly reducing the number of nodes expanded by the algorithm. This results in more benchmark instances of difficult optimization problems being solved in less time while using narrower decision diagrams. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0340 ), as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0340 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Discrete optimization problems expressible as dynamic programs can be solved by branch-and-bound with decision diagrams. This approach dynamically compiles bounded-width decision diagrams to derive both lower and upper bounds on unexplored parts of the search space, until they are all enumerated or discarded. Assuming a minimization problem, relaxed decision diagrams provide lower bounds through state merging while restricted decision diagrams obtain upper bounds by excluding states to limit their size. As the selection of states to merge or delete is done locally, it is very myopic to the global problem structure. In this paper, we propose a novel way to proceed that is based on pre-solving a so-called aggregate version of the problem with a limited number of states. The compiled decision diagram of this aggregate problem is tractable and can fit in memory. It can then be exploited by the original branch-and-bound to generate additional pruning and guide the compilation of restricted decision diagrams toward good solutions. The results of the numerical study we conducted on three combinatorial optimization problems show a clear improvement in the performance of DD-based solvers when blended with the proposed techniques. These results also suggest an approach where the aggregate dynamic programming model could be used in replacement of the relaxed decision diagrams altogether.

Local search is a popular technique to solve combinatorial optimization problems efficiently. To escape local minima one generally uses metaheuristics or try to design large neighborhoods around the current best solution. A somewhat more black box approach consists in using an optimization solver to explore a large neighborhood. This is the large-neighborhood search (LNS) idea that we reuse in this work. We introduce a generic neighborhood exploration algorithm based on restricted decision diagrams (DD) constructed from the current best solution. We experiment DD-LNS on two sequencing problems: the traveling salesman problem with time windows (TSPTW) and a production planning problem (DLSP). Despite its simplicity, DD-LNS is competitive with the state-of-the-art MIP approach on DLSP. It is able to improve the best known solutions of some standard instances for TSPTW and even to prove the optimality of quite a few other instances.

This paper presents ddo, a generic and efficient library to solve constraint optimization problems with decision diagrams. To that end, our framework implements the branch-and-bound approach which has recently been introduced by Bergman et al., (2016) to solve dynamic programs to optimality. Our library allowed us to successfully reproduce the results of Bergman et al. for MISP, MCP and MAX2SAT while using a single generic library. As an additional benefit, our ddo library is able to exploit parallel computing for its purpose without imposing any constraint on the user (apart from memory safety). Ddo is released as an open source rust library (crate) alongside with its companion example programs to solve the aforementioned problems. To the best of our knowledge, this is the first public implementation of a generic library to solve combinatorial optimization problems with branch-and-bound MDD.