Decision Diagram-Based Branch-and-Bound with Caching for Dominance and Suboptimality Detection
Vianney Coppé, Xavier Gillard, Pierre Schaus
TL;DR
The paper addresses the overhead of repeated work in decision diagram (DD)-based branch-and-bound (B&B) for discrete optimization modeled via dynamic programming (DP). It introduces a Cache of expansion thresholds θ(u|B̄) derived from relaxed DDs, combining dominance-based θ_d and pruning-based θ_p to decide when to expand DP states that share the same σ(u). This caching enables warm-starting and stronger pruning across successive approximate DDs, yielding substantial reductions in DD node expansions and faster solution times on challenging instances such as TSPTW, PSP, and SRFLP, sometimes surpassing state-of-the-art MIP performance. The approach is evaluated on three problems, demonstrates memory-time trade-offs, and discusses static vs dynamic variable orderings, with implications for parallelization and further improvements.
Abstract
The branch-and-bound algorithm based on decision diagrams introduced by Bergman et al. in 2016 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 the pruning inequalities of the filtering techniques introduced by Gillard et al. in 2021. Computational experiments show that the pruning brought by this caching mechanism allows 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.