Implicit Decision Diagrams

TL;DR

This paper introduces implicit Decision Diagrams that store arcs implicitly, reducing construction complexity from logarithmic or quadratic to linear per layer, and demonstrates their effectiveness in optimization problems like Subset Sum.

Contribution

The paper proposes a novel implicit DD data structure, proves its optimal complexity, and integrates it into a solver that outperforms Gurobi on specific benchmarks.

Findings

Implicit DDs reduce construction complexity to O(w) per layer.

Theoretical proof of optimality for the implicit DD framework.

Experimental results show improved performance over Gurobi on Subset Sum.

Abstract

Decision Diagrams (DDs) have emerged as a powerful tool for discrete optimization, with rapidly growing adoption. DDs are directed acyclic layered graphs; restricted DDs are a generalized greedy heuristic for finding feasible solutions, and relaxed DDs compute combinatorial relaxed bounds. There is substantial theory that leverages DD-based bounding, yet the complexity of constructing the DDs themselves has received little attention. Standard restricted DD construction requires $O(w \log(w))$ per layer; standard relaxed DD construction requires $O(w^2)$, where $w$ is the width of the DD. Increasing $w$ improves bound quality at the cost of more time and memory. We introduce implicit Decision Diagrams, storing arcs implicitly rather than explicitly, and reducing per-layer complexity to $O(w)$ for restricted and relaxed DDs. We prove this is optimal: any framework treating state-update and merge operations as black boxes cannot do better. Optimal complexity shifts the challenge from algorithmic overhead to low-level engineering. We show how implicit DDs can drive a MIP solver, and release ImplicitDDs, an open-source Julia solver exploiting the implementation refinements our theory enables. Experiments demonstrate the solver outperforms Gurobi on Subset Sum. Code (https://github.com/IsaacRudich/ImplicitDDs.jl)