Implicit Decision Diagrams

TL;DR

Implicit Decision Diagrams are introduced, storing arcs implicitly rather than explicitly, and reducing per-layer complexity to $O(w)$ for restricted and relaxed DDs, proving this is optimal: any framework treating state-update and merge operations as black boxes cannot do better.

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)

Figures (9)

• Figure 1: Exact decision diagram for $\mathcal{P}_{ex}$. X indicates a constraint violation.
• Figure 2: Restricted DD for $\mathcal{P}_{ex}$, $w\xspace = 3$. X marks infeasible or trimmed nodes.
• Figure 3: Relaxed DD for $\mathcal{P}_{ex}$, $w\xspace = 3$. Terminal $z$ is a lower bound.
• Figure 4: Implicit restricted DD for $\mathcal{P}_{ex}$ with $w\xspace=3$. Arcs below threshold $\tau$ are kept (dots); excess arcs are rejected (squares). The gray ticks on the number line indicate histogram bin edges. On $x_2$, one of four candidates is rejected. On $x_3$, one node is rejected arbitrarily from the bin above $\tau$, and two of the kept nodes are infeasible (X). Though visually leftmost (lowest objective), the rejected node is rightmost in memory order (highest parent index), so the budget expires before it is processed. Only one feasible node remains for $x_4$.
• Figure 5: Implicit relaxed DD construction for $\mathcal{P}_{ex}\xspace$ with $w\xspace=3$. The width-$K$ DD (left) is constructed first. Then, for each layer, we select thresholds (middle) and generate width-$w\xspace$ children (right). Left: Width-$K$ DD; each node shows state and inverted interval. Middle: Arcs placed on a number line by objective value; thresholds $j_{low}$ and $j_{high}$ determine which arcs are individualized versus merged. For $x_2$, there are four arcs at $z \in \{-5, -3, -2, 0\}$ with threshold $j_{low}=j_{high}=-3.3$: the $z=-5$ arc will be individualized, and the remaining arcs will be merged by label. Right: Width-$w\xspace$ DD; each node shows state, inverted interval, and objective value. For $x_2$, the $l=0$ node with interval $[1,2]$ is the merged node (the interval indicates it came from parents 1 and 2).

Theorems & Definitions (12)

• theorem thmcountertheorem: IP Exactness
• lemma thmcounterlemma
• proof
• proof : Theorem \ref{['thm:ip_exactness']}
• theorem thmcountertheorem: Tightness
• proof
• corollary thmcountercorollary: Optimality
• proof
• theorem thmcountertheorem: Feasibility Lower Bound
• proof