The complexity of semidefinite programs for testing $k$-block-positivity

TL;DR

This paper analyzes the computational complexity of testing $k$-block-positivity via semidefinite programs, revealing how symmetry reductions and representation theory influence the problem's difficulty and hierarchy collapse.

Contribution

It introduces a symmetry reduction scheme based on Young diagrams and derives an explicit complexity formula linked to irreducible representations of $erd)$, explaining hierarchy collapse at $k=d$.

Findings

Complexity linked to dimensions of irreducible representations.

Explicit formula for the complexity of $k$-block-positivity testing.

Hierarchy collapses when $k=d$ due to symmetry considerations.

Abstract

We extend \cite{chen2025srkbp} by analyzing the complexity of the $k$-block-positivity testing algorithm that stems from the optimization problem in Definition \ref{definition:SDP-k-block-positivity}. In this paper, we investigate a symmetry reduction scheme based on rectangular shaped Young diagrams. Connecting the complexity to the dimensions of irreducible representations of $\U(d)$, we derive an explicit formula for the complexity, which also clarifies why the semidefinite program hierarchy collapses in the $k=d$ case.