Improved Peel-and-Bound: Methods for Generating Dual Bounds with Multivalued Decision Diagrams

TL;DR

This paper enhances decision diagram techniques for discrete optimization by introducing peel-and-bound, which improves dual bounds efficiently and outperforms existing methods on complex sequencing problems.

Contribution

The paper presents novel peel-and-bound methods, including hyper-optimization and parallelization strategies, extending decision diagram applications to a broader class of problems.

Findings

Peel-and-bound generates stronger bounds than branch-and-bound with the same propagators.

The method outperforms ddo on the TSPTW, closing 15 benchmark instances.

New algorithms are generalized for any discrete optimization problem.

Abstract

Decision diagrams are an increasingly important tool in cutting-edge solvers for discrete optimization. However, the field of decision diagrams is relatively new, and is still incorporating the library of techniques that conventional solvers have had decades to build. We drew inspiration from the warm-start technique used in conventional solvers to address one of the major challenges faced by decision diagram based methods. Decision diagrams become more useful the wider they are allowed to be, but also become more costly to generate, especially with large numbers of variables. In the original version of this paper, we presented a method of peeling off a sub-graph of previously constructed diagrams and using it as the initial diagram for subsequent iterations that we call peel-and-bound. We tested the method on the sequence ordering problem, and our results indicate that our peel-and-bound scheme generates stronger bounds than a branch-and-bound scheme using the same propagators, and at significantly less computational cost. In this extended version of the paper, we also propose new methods for using relaxed decision diagrams to improve the solutions found using restricted decision diagrams, discuss the heuristic decisions involved with the parallelization of peel-and-bound, and discuss how peel-and-bound can be hyper-optimized for sequencing problems. Furthermore, we test the new methods on the sequence ordering problem and the traveling salesman problem with time-windows (TSPTW), and include an updated and generalized implementation of the algorithm capable of handling any discrete optimization problem. The new results show that peel-and-bound outperforms ddo (a decision diagram based branch-and-bound solver) on the TSPTW. We also close 15 open benchmark instances of the TSPTW.