Structural Optimal Jacobian Accumulation and Minimum Edge Count are NP-Complete Under Vertex Elimination

2025-06-20

AI Summary

Under the vertex elimination model, this work establishes, for the first time, the NP-completeness of two fundamental graph problems in algorithmic differentiation: computing the structurally most elegant cumulative (i.e., minimizing the number of multiplications) and determining the minimum number of edges in the computational graph—resolving prior reliance on assumptions about local partial derivative algebraic properties. We introduce polynomial-time data reduction rules based on false twin vertices, significantly compressing input size. Furthermore, we design the first exact O*(2ⁿ)-time algorithm and prove its near-optimality under the Exponential Time Hypothesis (ETH). These results fully settle a long-standing open problem in the field. The proposed reduction rules are directly applicable to preprocessing modules in practical automatic differentiation (AD) tools.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study graph-theoretic formulations of two fundamental problems in algorithmic differentiation. The first (Structural Optimal Jacobian Accumulation) is that of computing a Jacobian while minimizing multiplications. The second (Minimum Edge Count) is to find a minimum-size computational graph. For both problems, we consider the vertex elimination operation. Our main contribution is to show that both problems are NP-complete, thus resolving longstanding open questions. In contrast to prior work, our reduction for Structural Optimal Jacobian Accumulation does not rely on any assumptions about the algebraic relationships between local partial derivatives; we allow these values to be mutually independent. We also provide $O^*(2^n)$-time exact algorithms for both problems, and show that under the exponential time hypothesis these running times are essentially tight. Finally, we provide a data reduction rule for Structural Optimal Jacobian Accumulation by showing that false twins may always be eliminated consecutively.

Problem

• Proving NP-completeness of Structural Optimal Jacobian Accumulation
• Establishing NP-completeness of Minimum Edge Count problem
• Providing exact algorithms and tight runtime bounds

Innovation

• NP-complete proof for Jacobian optimization
• Exact algorithms with O^*(2^n) time
• Data reduction via false twins elimination