An efficient search strategy for hidden ideals in pointed partially ordered sets

TL;DR

This paper introduces an efficient search strategy for identifying unknown ideals in pointed posets with limited queries, providing bounds and asymptotic optimality results, especially for complete -ary trees.

Contribution

It develops a general search strategy for hidden ideals in pointed posets and establishes new bounds on query complexity, demonstrating asymptotic optimality in specific tree structures.

Findings

Provides bounds on the number of queries needed to find an ideal

Strategy performs asymptotically optimally on complete -ary trees

Offers a general approach applicable to various poset structures

Abstract

We consider a combinatorial question about searching for an unknown ideal $\mu$ within a known pointed poset $\lambda$. Elements of $\lambda$ may be queried for membership in $\mu$, but at most $k$ positive queries are permitted. We provide a general search strategy for this problem, and establish new bounds (based on $k$ and the degree and height of $\lambda$) for the total number of queries required to identify $\mu$. We show that this strategy performs asymptotically optimally on the family of complete $\ell$-ary trees as the height grows.