Testing idealness in the filter oracle model

TL;DR

This paper investigates the computational complexity of testing whether a clutter is ideal using a filter oracle, establishing an exponential lower bound on the number of oracle calls needed in the worst case.

Contribution

It proves a tight exponential lower bound on the number of filter oracle calls required to test idealness of a clutter, using the theory of cuboids.

Findings

Any algorithm must make at least 2^n calls in the worst case.

The proof employs the theory of cuboids to establish the bound.

The result highlights the inherent complexity of the problem.

Abstract

A filter oracle for a clutter consists of a finite set $V$ along with an oracle which, given any set $X\subseteq V$, decides in unit time whether or not $X$ contains a member of the clutter. Let $\mathfrak{A}_{2n}$ be an algorithm that, given any clutter $\mathcal{C}$ over $2n$ elements via a filter oracle, decides whether or not $\mathcal{C}$ is ideal. We prove that in the worst case, $\mathfrak{A}_{2n}$ must make at least $2^n$ calls to the filter oracle. Our proof uses the theory of cuboids.