Learning Partitions using Rank Queries

TL;DR

The paper addresses learning an unknown partition of an $n$-element universe using rank queries that return the number of parts hit by a subset. It introduces a simple deterministic $O(n)$-query algorithm that builds a collection of independent sets, uses coin-weighing-inspired Merge operations to identify common elements, and reconstructs the partition from a representative mapping. It generalizes to general partition matroids with $O(n+k\log r)$ queries by reducing to the simple partition case via finding two representative sets, where $r=\sum_i r_i$. The results bridge combinatorial search, matroid learning, and hypergraph reconstruction under rank queries, and they raise open questions about optimal constants and extending the approach to dense partition structures.

Abstract

We consider the problem of learning an unknown partition of an $n$ element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple $O(n)$-query, efficient, deterministic algorithm for this problem. We also generalize to give an $O(n + k\log r)$-rank query algorithm for a general partition matroid where $k$ is the number of parts and $r$ is the rank of the matroid.

Figures (2)

• Figure 1: After we merge $I_1, I_2$ on the left, we get $I_3$ and a mapping from $\mathsf{com}(I_1, I_2)$ to $\mathsf{com}(I_2, I_1)$.
• Figure 2: Illustration of how we simulate a simple partition matroid rank query inside of a basis with representatives outside. We have a basis with $b_i$s equal to $1$ (purple nodes), $2$ (green), $3$ (blue) and $4$ (black) respectively. $\mathsf{rank}(B) = 10$, $\mathsf{rank}(B- S) = 5$. Note $|B-S+T_2| = 10$, yet $\mathsf{rank}(B-S+T_2) = 9$ because the number of green nodes is capped at 2. $\mathsf{rank}(B-S+T_2) - \mathsf{rank}(B-S) = 3$ simulate a simple partition matroid rank query for $S$. The circled nodes correspond to the 3 partitions included in $S$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1: Bshouty09
• Lemma 2
• Lemma 3
• proof