Implicit Decision Diagrams
Isaac Rudich, Louis-Martin Rousseau

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 per layer; standard relaxed DD construction requires , where is the width of the DD. Increasing 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 for restricted and relaxed DDs. We prove this is optimal: any framework treating state-update…
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Taxonomy
TopicsConstraint Satisfaction and Optimization · Complexity and Algorithms in Graphs · Formal Methods in Verification
