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

TL;DR

This paper proves that both the Structural Optimal Jacobian Accumulation and Minimum Edge Count problems are NP-complete when using vertex elimination, providing exact algorithms and data reduction rules.

Contribution

It establishes NP-completeness for both problems without algebraic assumptions and offers $O^*(2^n)$ algorithms along with a data reduction rule for false twins.

Findings

Both problems are NP-complete.

Exact $O^*(2^n)$ algorithms are provided.

False twins can be eliminated through data reduction.

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.