Prove Symbolic Regression is NP-hard by Symbol Graph

TL;DR

This paper demonstrates that symbolic regression is NP-hard by introducing the concept of symbol graphs and linking the problem to the NP-hard Steiner Arborescence problem.

Contribution

It introduces symbol graphs as a novel representation and proves the NP-hardness of symbolic regression through a reduction to a known NP-hard problem.

Findings

Symbol graphs effectively represent the entire expression space.

Symbolic regression is NP-hard due to its connection with DCSAP.

The complexity of SR is established through reduction to Steiner Arborescence problem.

Abstract

Symbolic regression (SR) is the task of discovering a symbolic expression that fits a given data set from the space of mathematical expressions. Despite the abundance of research surrounding the SR problem, there's a scarcity of works that confirm its NP-hard nature. Therefore, this paper introduces the concept of a symbol graph as a comprehensive representation of the entire mathematical expression space, effectively illustrating the NP-hard characteristics of the SR problem. Leveraging the symbol graph, we establish a connection between the SR problem and the task of identifying an optimally fitted degree-constrained Steiner Arborescence (DCSAP). The complexity of DCSAP, which is proven to be NP-hard, directly implies the NP-hard nature of the SR problem.